Markov convexity and local rigidity of distorted metrics

Mark o v con v exit y a nd l o cal rigidity of distor ted metrics Manor Mendel ∗ Op en Univ ersit y of Israel manorme@ope nu.ac.il Assaf Naor † New Y ork Univ ersit y naor@cims.n yu.edu Abstract It is shown that a Banach space admits an equiv a lent norm whose mo dulus of uniform con- vexit y has pow er-type p if and only if it is Markov p -conv ex. Counterexamples ar e co nstructed to natural questions related to iso morphic uniform conv e x ity of metr ic spa ces, showing in particular that tree metr ics fail to have the dichotom y prop er ty . Con ten ts 1 In tro duction 2 1.1 The n on existence of a metric dichoto m y for trees . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Ov erview of the p ro ofs of T heorem 1.10 and Theorem 1.12 . . . . . . . . . . 7 2 Mark ov p -co n v exit y and p -con v exity coincide 9 3 A doubling space whic h is not Mark ov p -conv ex for any p ∈ (0 , ∞ ) 15 4 Lipsc hitz quotients 17 5 A dic hotom y theorem for vertically faithful em b eddings of trees 18 6 T ree metrics do not ha v e the dic hotom y prop erty 21 6.1 Horizon tally con tracted trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.2 Geometry of H -trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.2.1 Classification of ap p ro ximate midp oint s . . . . . . . . . . . . . . . . . . . . . 23 6.2.2 Classification of ap p ro ximate forks . . . . . . . . . . . . . . . . . . . . . . . . 30 6.2.3 Classification of ap p ro ximate 3-paths . . . . . . . . . . . . . . . . . . . . . . . 36 6.3 Nonem b eddability of vertica lly faithful B 4 . . . . . . . . . . . . . . . . . . . . . . . . 42 6.4 Nonem b eddability of b inary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7 Discussion and op en problems 45 ∗ Supp orted by ISF grant 221/0 7, BSF grant 2006009 , and a gift from Cisco researc h center. † Supp orted by NSF gran ts CCF-0635 078 and CCF-0832795, BSF gran t 2006009 , and th e P ack ard F oundation. 1 1 In tro duction A Banac h space ( X , k · k X ) is said to b e fin itely repr esen table in a Banac h sp ace ( Y , k · k Y ) if th ere exists a constan t D < ∞ such that for every finite d im en sional linear subs p ace F ⊆ X t here is a linear op erator T : F → Y sa tisfying k x k X 6 k T x k Y 6 D k x k X for all x ∈ F . In 1976 Rib e [31] pro v ed that if t wo Banac h spaces X and Y are uniform ly h omeomorphic, i.e., there is a bijection f : X → Y suc h that f and f − 1 are uniformly con tin uous, then X is fin itely representa ble in Y and vice versa. Th is remark able theorem motiv ated wh at is known to day as the “Rib e program”: the searc h for pur ely metric reformula tions of basic linear concepts an d in v arian ts fr om the lo cal theory of Banac h spaces. T his researc h program was put f orth b y Bourgain in 1986 [5]. Since its inception, th e Rib e program attracted the wo rk of many m athematicians, and led to the dev elopmen t of sev eral satisfactory metric theories that extend imp ortant concepts and r esults of Banac h space theory; see the introd uction of [24] for a historical discussion. So far, pr ogress on the Rib e program has come hand -in-hand with applications to metric geometry , group theory , functional analysis, and computer science. The presen t pap er contai ns further progress in this direction: we obtain a metric charac terization of p -conv exit y in Banac h spaces, derive some of its metric consequences, and construct unexp ected counter-exa mples which indicate that furth er progress on the Rib e program can unco ver n onlinear phenomena that are mark edly d ifferen t fr om their Banac h sp ace counterparts. In doing so, we answer questions p osed by Lee-Naor-P eres and F efferman, and impro v e a theorem of Bates, Johnson, Linden strauss, P reiss and Sc hec htman. Th ese results, wh ic h will b e explained in d etail b elo w, w ere ann ounced in [23]. F or p > 2, a Banach space ( X , k · k X ) is said to b e p -conv ex if there exists a norm | | | · | | | which is equiv alen t to k · k X (i.e., for some a, b > 0, a k x k X 6 | | | x | | | 6 b k x k X for all x ∈ X ), and a constan t K > 0 satisfying: | | | x | | | = | | | y | | | = 1 = ⇒             x + y 2             6 1 − K | | | x − y | | | p . (1) X is called sup err eflexiv e if it is p -c on v ex for some p > 2 (historically , this is n ot the original definition of sup errefl exivit y 1 , but it is equ iv alen t to it due to a deep theorem of P isier [29], wh ich builds on imp ortan t work of James [10] and Enflo [7]). F or concreteness, w e recall (see, e.g., [2]) that L p is 2-con vex for p ∈ (1 , 2] and p -con v ex for p ∈ [2 , ∞ ). Rib e’s theorem implies that p -conv exit y , and hence also sup erreflexivit y , is preserv ed under uniform homeomorphisms . The fir st ma jor su ccess of the Rib e program is a famous theorem of Bourgain [5] whic h obtains a metrical c haracterization of su p err efl exivit y as f ollo ws. Theorem 1.1 (Bourgain’s m etrical characte rization of sup erreflexivit y [5]) . L et B n b e the c omplete unweighte d binary tr e e of depth n , e quipp e d with the nat ur al gr aph-the or etic al metric. Then a Banach sp ac e X is sup err eflexive if and only if lim n →∞ c X ( B n ) = ∞ . (2) Here, and in what follo ws , giv en t wo m etric spaces ( M , d M ), ( N , d N ), the parameter c M ( N ) denotes the smallest bi-Lipsc hitz distortion with whic h N em b eds in to M , i.e., the infim um of 1 James’ original definition of sup erreflexivity is that a Banac h space X is superrefl ex ive if “its lo cal structure forces reflexivity”, i.e., if every Banach space Y t h at is finitely represen table in X must b e reflexive. Enflo’s renorming theorem states that sup erreflex ivity is equiv alent to having an equ iv alent norm ||| · ||| that is uniformly convex, i.e., for every ε ∈ (0 , 1) there exists δ > 0 suc h th at if ||| x ||| = ||| y ||| = 1 and ||| x − y ||| = ε then ||| x + y ||| 6 2 − δ . 2 those D > 0 suc h that there exists a scaling factor r > 0 and a mapping f : N → M satisfying r d N ( x, y ) 6 d M ( x, y ) 6 D r d N ( x, y ) for all x, y ∈ N (if n o suc h f exists then set c M ( N ) = ∞ ). Bourgain’s theorem characte rizes sup err eflexivit y of Banac h spaces in terms of their metric structure, b ut it lea v es op en th e c haracterization of p -con vexit y . The notion of p -con v exit y is crucial for many applicatio ns in Banac h sp ace th eory and m etric geometry , and it turns out that the completion of the Rib e pr ogram f or p -con v exit y r equires significan t additional work b ey ond Bourgain’s su p erreflexivity theorem. As a firs t step in this direction, Lee, Naor an d Pe res [16] defined a b i-Lipsc hitz inv ariant of metric sp aces called Markov c onvexity , whic h is motiv ated by Ball’s n otion of Mark o v t yp e [1] and Bourgain’s argument in [5]. Definition 1.2 ([16]) . L et { X t } t ∈ Z b e a Markov chain on a state sp ac e Ω . Given an inte ger k > 0 , we denote by { e X t ( k ) } t ∈ Z the pr o c ess which e quals X t for time t 6 k , and evolves indep endently (with r esp e ct to the sam e tr ansition pr ob abilities) for time t > k . Fix p > 0 . A metric sp ac e ( X, d X ) is c al le d Markov p -c onvex with c onstant Π if for e very M arkov chain { X t } t ∈ Z on a state sp ac e Ω , and every f : Ω → X , ∞ X k =0 X t ∈ Z E h d X  f ( X t ) , f  e X t  t − 2 k   p i 2 k p 6 Π p · X t ∈ Z E  d X ( f ( X t ) , f ( X t − 1 )) p  . (3) The le ast c onstant Π for which (3) holds for al l Markov chains is c al le d the Markov p -c onvexity c onstant of X , and is denote d Π p ( X ) . We shal l say that ( X , d X ) is M arkov p -c onvex if Π p ( X ) < ∞ . T o gain intuition for Definition 1.2, consid er the standard d o w n wa rd r andom w alk starting f rom the ro ot of th e binary tree B n (with absorbin g states at th e lea v es). F or an arb itrary mappin g f from B n to a metric space ( X, d X ), the triangle inequalit y implies that for eac h k ∈ N we ha v e X t ∈ Z E h d X  f ( X t ) , f  e X t  t − 2 k   p i 2 k p . p X t ∈ Z E  d X ( f ( X t ) , f ( X t − 1 )) p  , (4) with asymp totic equalit y (up to constan ts dep ending only on p ) for k 6 log n 2 when X = B n and f is the identit y map p ing. On the other hand, if X is a Mark o v p -con vex space then the sum o ver k of the left-hand side of (4) is uniformly b ounde d by the righ t-hand side of (4), and therefore Mark o v p -con vex sp aces cannot con tain B n with distortion uniformly b ounded in n . W e r efer to [16] for more in f ormation on the notion of Marko v p -con v exit y . In particular, it is sho wn in [16] that th e Marko v 2-conv exit y constan t of an arbitrary weigh ted tree T is, up to constan t factors, the Eu clidean distortion of T . W e refer to [16] for L p v ersions of this statemen t and their algorithmic applications. It wa s also sho wn in [16], via a mo dification of an argumen t of Bourgain [5], that if a Banac h space X is p -con v ex then it is also Marko v p -conv ex. It w as aske d in [16] if the con verse is also true. Here we answ er th is question p ositiv ely: Theorem 1.3. A Banach sp ac e is p -c onvex if and only if it is M arkov p -c onvex. Th us Mark o v p -conv exit y is equiv alen t to p -conv exit y in Banac h spaces, completing the Rib e program in this case. Our pro of of Theorem 1.3 is based on a r enorming metho d of Pisier [29]. It can b e viewed as a n onlinear v arian t of Pisier’s argu m en t, and seve ral subtle c hanges are required in order to adapt it to a n onlinear condition such as (3). 3 Results similar to Theorem 1.3 hav e b een obtained for th e notions of type an d cot yp e of Banac h spaces (see [6, 30, 1, 25, 24, 22]), and h a ve b een used to tr an s fer some of the linear theory to the setting of general metric sp aces. This led to several applications to problems in metric geometry . Apart from the applications of Mark o v p -con vexit y that w ere obtained in [16], h er e we sho w that this in v arian t is pr eserv ed under Lipsc hitz quotient s. T he notion of Lipsc hitz quotien t w as int ro du ced b y Gromo v [8, Sec. 1.25]. Given t wo metric spaces ( X , d X ) and ( Y , d Y ), a s urjectiv e mappin g f : X → Y is called a L ipsc hitz quotien t if it is Lip sc h itz, and it is also “Lipschitzl y op en” in the sense that there exists a constant c > 0 su ch that for ev ery x ∈ X and r > 0, f ( B X ( x, r )) ⊇ B Y  f ( x ) , r c  . (5) Here we sho w the follo w in g r esult: Theorem 1.4. If ( X , d X ) is Markov p - c onvex and ( Y , d Y ) is a Lipschitz quotient of X , then Y is also Markov p -c onvex. In [3] Bates, Johnson, Lindens tr auss, Preiss and Schec h tman inv estigated in detail Lip sc h itz quotien ts o f Banac h spaces. Their results imply that if 2 6 p < q then L q is not a Lip sc h itz quotien t of L p . Since L p is p -con v ex, it is also Mark o v p -con v ex. Hence also all of its subsets are Mark ov p conv ex. But, L q is not p -con vex, so w e dedu ce that L q is not a Lip s c h itz quotien t of any subset of L p . Thus our new “inv ariant approac h” to the ab ov e result of [3] significan tly extends it. Note that the method of [3] is based on a differen tiation argumen t, and h ence it crucially relies on the fact that the L ipsc hitz quotien t mapp ing is defined on all of L p and not ju st on an arbitrary subset of L p . In ligh t of T heorem 1.3 it is n atural to ask if Bourgain’s c haracterizatio n of sup erreflexivit y holds for general metric spaces. Namely , is it true that for an y metric space X , if lim n →∞ c X ( B n ) = ∞ then X is Marko v p -con vex for some p < ∞ ? This qu estion was asked in [16]. Here w e sho w that the ans w er is negativ e: Theorem 1.5. Ther e exists a metric sp ac e ( X, d X ) which is not Markov p -c onvex for any p ∈ (0 , ∞ ) , yet lim n →∞ c X ( B n ) = ∞ . In fact, ( X , d X ) c an b e a doubling metric sp ac e, and henc e c X ( B n ) > 2 κn for some c onstant κ > 0 . Theorem 1.5 is in sharp con trast to the previously established metric c haracterizations of the linear notions of type and cot yp e. Sp ecifically , it w as sho wn by Bourgain, Milman an d W olfson [6] that any metric space with n o nontrivial metric t yp e m ust conta in the Hamming cub es ( { 0 , 1 } n , k ·k 1 ) with distortion indep endent of n . An analogous result was obtained in [24 ] for metric spaces with no nontrivial metric cot yp e, with the Hamming cub e replaced by the ℓ ∞ grid ( { 1 , . . . , m } n , k · k ∞ ). Our pro of of Th eorem 1.5 is based on an analysis of the b eha vior of a certain Marko v chain on the Laakso graph s: a sequen ce of com binatorial graphs whose d efinition is recalled in Section 3. As a consequence of this analysis, we obtain the f ollo wing distortion lo we r b ound : Theorem 1.6. F or any p > 2 , the L aakso gr aph of c ar dinality n incurs distortio n Ω  (log n ) 1 /p  in any emb e dding into a p -c onvex Banach sp ac e. Th us, in particular, for p > 2 the n -p oin t Laakso graph incur s distortion Ω  (log n ) 1 /p  in any em b eddin g in to L p . Th e case of L p em b eddin gs of the Laakso graphs when 1 < p 6 2 w as already solv ed i n [26, 12, 15, 14] using the un iform 2-con v exit y prop erty of L p . But, these pro ofs rely 4 crucially on 2-con vexit y and do not extend to the case of p -con vexit y when p > 2. Sub sequen t to the p ublication of our pro of of T heorem 1.6 in the announcemen t [23], an alternative p r o of of this fact was recen tly disco vered by Johnson and Sc hec htman in [11]. 1.1 The nonexistence of a metric dic hotomy for trees Bourgain’s metrical c h aracterizati on of sup erreflexivit y yields the follo wing statemen t: Theorem 1.7 (Bourgain’s tree dichoto m y [5]) . F or any Banach sp ac e ( X , k·k X ) one of the fol lowing two dichotomic p ossibilities must hold true: • either for al l n ∈ N we have c X ( B n ) = 1 , • or ther e exists α = α X > 0 such that for al l n ∈ N we have c X ( B n ) > (log n ) α . Th us, ther e is a gap in the p ossible rates of gro w th of the sequence { c X ( B n ) } ∞ n =1 when X is a Banac h space; consequently , if w e w ere told that, sa y , c X ( B n ) = O (log log n ), th en we would immediately d ed uce that actually c X ( B n ) = 1 for all n . Additional gap results of this t yp e are kno wn d ue to the theory of nonlinear t yp e and cot yp e: Theorem 1.8 (Bourgain-Milman-W ol fson cub e dic hotom y [6]) . F or any metric sp ac e ( X , d X ) one of the fol lowing two dichotomic p ossibilities must hold true: • either for al l n ∈ N we have c X ( { 0 , 1 } n , k · k 1 ) = 1 , • or ther e exists α = α X > 0 such that for al l n ∈ N we have c X ( { 0 , 1 } n , k · k 1 ) > n α . Theorem 1.8 is a m etric analog ue of Pisier’s charac terization [28] of Banac h sp aces with trivial Rademac her t yp e. A metric analog ue of the Maurey-Pisier c haracterization [20] of Banac h spaces with fin ite Rademac h er cot yp e yields the follo wing dic hotom y result for ℓ ∞ grids: Theorem 1.9 (Grid dic hotom y [24]) . F or any metric sp ac e ( X , d X ) one of the fol lowing two dichotomic p ossibilities must hold true: • either for al l n ∈ N we have c X ( { 0 , . . . , n } n , k · k ∞ ) = 1 , • or ther e exists α = α X > 0 such that for al l n ∈ N we have c X ( { 0 , . . . , n } n , k · k ∞ ) > n α . W e refer to th e survey article [21] for more information on the theory of metric dic h otomies. Note that Theorem 1.7 is stated for Banac h spaces, wh ile Theorem 1.8 and Theorem 1.9 h old for general metric sp aces. One might exp ect that as in the case of previous progress on Rib e’s program, a m etric theory of p -con v exity w ould result in a p r o of that Th eorem 1.7 holds when X is a general metric space. Sur prisingly , we sho w h er e that this is not tru e: Theorem 1.10. Ther e exists a universal c onstant C > 0 with the fol lowing pr op erty. Assume that { s ( n ) } ∞ n =0 ⊆ [4 , ∞ ) is a nonde cr e asing se quenc e such that { n/s ( n ) } ∞ n =0 is also nonde cr e asing. Then ther e exists a metric sp ac e ( X , d X ) satisfying for al l n > 2 , s  n 40 s ( n )   1 − C s ( n ) log s ( n ) log n  6 c X ( B n ) 6 s ( n ) . (6) Thus, assuming that s ( n ) = o  log n log log n  , ther e exists a subse quenc e { n k } ∞ k =1 for which (1 − o (1)) s ( n k ) 6 c X ( B n k ) 6 s ( n k ) . (7) 5 Theorem 1.10 sho ws that unlike the case of Banac h spaces, f or general metric sp aces, c X ( B n ) can h a ve an arbitrarily slo w gro wth rate. Bourgain, Milman and W olfson also obtained in [6] the follo wing finitary version of Th eorem 1.8: Theorem 1.11 (Lo cal rigidit y of Hamming cub es [6]) . F or every ε > 0 , D > 1 and n ∈ N ther e exists m = m ( ε, D , n ) ∈ N such that lim n →∞ m ( ε, D , n ) = ∞ , and for every metric d on { 0 , 1 } n which is bi-Lipschitz with distortion 6 D to the ℓ 1 (Hamming) metric, c ( { 0 , 1 } n ,d ) ( { 0 , 1 } m , k · k 1 ) 6 1 + ε. W e refer to [6] (see also [30]) for b oun ds on m ( ε, D, n ). I n formally , Th eorem 1.11 says that the Hamming cub e ( { 0 , 1 } n , k · k 1 ) is lo c al ly rigid in the follo wing s en se: it is imp ossible to d istort the Hamming metric on a su fficien tly large hypercub e without th e resu lting m etric sp ace conta ining a hardly d istorted cop y of an arb itrarily large Hamming cub e. Stated in this w a y , Theorem 1.11 is a metric version of James’ theorem [9] that ℓ 1 is not a disto rtable space. The analogue of Theorem 1.11 with the Hamming cub e replaced b y the ℓ ∞ grid ( { 0 , . . . , n } n , k · k ∞ ) is Matou ˇ sek’s BD-Ramsey theorem [19]; see [24] for quan titativ e resu lts of th is t yp e in the ℓ ∞ case. The follo w in g v arian t of Th eorem 1.10 sh o w s that a lo cal r igidit y statemen t as ab o v e fails to hold tru e for binary trees; it can also b e view ed as a negativ e solution of the d istortion problem f or the in finite binary tree (see [27] and [4, C h. 13, 14] for more information on the d istortion pr ob lem for Banac h spaces). Theorem 1.12. L et B ∞ b e the c omplete unweighte d infinite binary tr e e. F or every D > 4 ther e exists a metric d on B ∞ that is D -e quivalent to the original shortest-p ath metric on B ∞ , yet f or every ε ∈ (0 , 1) and m ∈ N , c ( B ∞ ,d ) ( B m ) 6 D − ε = ⇒ m 6 D C D 2 /ε . The lo cal r igidit y problem for b in ary tr ees was stud ied by sev eral mathematicians. In p artic- ular, C . F efferman ask ed (priv ate comm u nication, 2005) wh ether { B n } ∞ n =1 ha v e the lo cal rigidit y prop erty , and Theorem 1.12 answers this question n egativ ely . F efferman also p ro v ed a partial lo cal rigidit y resu lt w hic h is a non-quan titativ e v ariant of Theorem 1.14 b elo w (see also Section 5). W e are v ery grateful to C. F efferman for asking u s the qu estion that led to the coun ter-examples of Theorem 1.10 and Theorem 1.12, for sharing with us his partial p ositive r esults, and f or encouraging us to work on these q u estions. M. Gromo v also in v estigated the lo cal r igidit y prob lem for bin ary trees, an d p ro v ed (via d ifferen t metho ds) n on-quant itativ e partial p ositiv e r esults in the spir it of Theorem 1.14. W e thank M. Gromo v for sharing with us his unp ublished wo rk on this topic. The results of T heorem 1.10 and Theorem 1.12 are qu ite unexp ected. Unfortu n ately , their pr o ofs are delicate and lengthy , and as such constitute the most in vol v ed p art of this article . In order to facilitat e the un derstanding of these constructions, we end the in tro du ction with an o verview of the main geometric ideas th at are used in their pro ofs. This is done in Section 1.1.1 b elo w —w e recommend reading this section first b efore delving into the tec h nical details presented in S ection 6. 6 1.1.1 Ov erview of t he pro ofs of Theorem 1.10 and Theorem 1.12 F or x ∈ B ∞ let h ( x ) b e its dep th , i.e., its distance from th e ro ot. Also , for x, y ∈ B ∞ let lca ( x, y ) denote their least common ancestor. The tree metric on B ∞ is then giv en b y: d B ∞ ( x, y ) = h ( x ) + h ( y ) − 2 h ( lca ( x, y )) . The metric space X o f Theorem 1.10 will b e B ∞ as a set, with a new m etric defin ed as f ollo ws. Giv en a sequence ε = { ε n } ∞ n =0 ⊆ (0 , 1] we define d ε : B ∞ × B ∞ → [0 , ∞ ) by d ε ( x, y ) = | h ( y ) − h ( x ) | + 2 ε min { h ( x ) ,h ( y ) } · [m in { h ( x ) , h ( y ) } − h ( lca ( x, y ))] . d ε do es not necessarily satisfy the triangle inequalit y , but und er some simple conditions on the sequence { ε n } ∞ n =0 it do es b ecome a metric on B ∞ ; see Lemma 6.1. A p ictorial descrip tion of the metric d ε is contai ned in Figure 1. No te that when ε n = 1 for all n , we h av e d ε = d B ∞ . Belo w we call th e metric spaces ( B ∞ , d ε ) horizon tally distorted trees, or H -trees, in short. lca( x, y ) x y b a ro ot d ε ( x, y ) = b + 2 a · ε h ( x ) Figure 1: The metric d ε define d on B ∞ . The arr ows indic ate horizonta l c ontr action by ε h ( x ) . The metric space ( X , d X ) of Theorem 1.10 will b e ( B ∞ , d ε ), where ε n = 1 /s ( n ) for all n . The iden tit y m apping of B n in to the top n -lev els of B ∞ has d istortion at most s ( n ), and therefore c X ( B n ) 6 s ( n ). T he c hallenge is to pro v e the lo wer b ound on c X ( B n ) in (6). Ou r initial approac h to lo wer-b ounding c X ( B n ) w as Matou ˇ sek’s metric differen tiation p r o of [18] of asymptotically sharp distortion low er b oun ds for embedd ings of B n in to un iformly conv ex Banac h spaces. F ollo wing Matou ˇ sek’s terminology [18], for δ > 0 a quadru ple of p oin ts ( x, y , z , w ) in a met- ric space ( X, d X ) is call ed a δ -fork if y ∈ Mid( x, z , δ ) ∩ Mid( x, w , δ ), where for a, b ∈ X the set of δ -appro ximate midp oin ts Mid( a, b, δ ) ⊆ X is defin ed as the s et of all w ∈ X satisfying max { d X ( x, y ) , d X ( y , z ) } 6 1+ δ 2 · d X ( x, z ). Th e p oints z , w will b e called b elo w the prongs of th e δ -fork ( x, y , z , w ). Ma tou ˇ sek starts with the observ ation that if X is a unif orm ly conv ex Banac h space then in any δ -fork in X the distance b et ween the prongs must b e muc h smaller (as δ → 0) than d X ( x, y ). Matou ˇ sek then sh o w s that for all D > 0, any d istortion D embedd ing of B n in to X m ust map some 0-fork in B n to a δ -fork in X , pro vid ed n is large enough (as a function of D and δ ). This reasoning immediately implies that c X ( B n ) m u st b e large when X is a uniformly con v ex Banac h space, and a clev er argument of Matou ˇ sek in [18] tu r ns this qu alitativ e argumen t into sh arp quan titativ e b ounds. 7 Of cour se, we cann ot hop e to use the ab o ve argumen t of Matou ˇ sek in order to pro v e Theo- rem 1.10, since Bourgain’s tree d ic hotomy theorem (T h eorem 1.7) d o es hold true for Banach sp aces. But, p erhaps w e can mimic this u niform conv exit y argumen t for other target m etric sp aces? On the face of it, H -trees are ideally suited f or this pu r p ose, since the horizon tal con tractions that we in tro duced shrink distances b et w een the pr on gs of canonical forks (call ( x, y , z , w ) ∈ B ∞ a canonical fork if x is an ancestor of y and z , w are descendan ts of y at depth h ( x ) + 2( h ( y ) − h ( x ))). It is for this reason exactly that w e d efined H -trees. Unfortunately , the situati on isn’t so simple. It turns out that H -trees do n ot b eha v e lik e uniformly con v ex Banac h spaces in terms of the prong-con tractions that they im p ose of δ -forks. H -trees can even con tain larger problematic configurations that h a ve sev eral und istorted δ -forks; suc h an example is depicted in Figure 2. r h 2 − h 1 = δ − 1 h 1 h 2 r Figure 2: The metric sp ac e on the right is the H -tr e e ( B ∞ , d ε ) , wher e ε n = δ for al l n . The pictur e describ es an e mb e dding of the tr e e on the left ( B 3 minus 4 le aves) into ( B ∞ , d ε ) with distortion at most 6 , yet al l anc estor/desc endant distanc es ar e distorte d by at most 1 + O ( δ ) . Th us, in ord er to pro v e Theorem 1.10 it do es not s u ffice to use Matou ˇ sek’s argumen t that a bi-Lipsc hitz em b edding of a large enough B n m ust send some 0-fork to a δ -fork. B ut, it turns out that this argumen t app lies n ot only to forks, b ut also to larger configurations. Definition 1.13. L et ( T , d T ) b e a tr e e with r o ot r , and let ( X , d X ) b e a metric sp ac e. A map ping f : T → X is c al le d a D - vertic al ly faith ful emb e dding if ther e exists a (sc aling factor) λ > 0 satisfying for any x, y ∈ T such that x is an anc estor of y , λd T ( x, y ) 6 d X ( f ( x ) , f ( y )) 6 D λd T ( x, y ) . (8) Recall that the d istortion of a mapping φ : M → N b et ween m etric spaces ( M , d M ) and ( N , d N ) is defined as dist( φ ) def =    sup x,y ∈ M x 6 = y d N ( φ ( x ) , φ ( y )) d M ( x, y )    ·    sup x,y ∈ M x 6 = y d M ( x, y ) d N ( φ ( x ) , φ ( y ))    ∈ [1 , ∞ ] . With this terminology , w e can state the follo win g crucial result. 8 Theorem 1.14. Ther e exists a universal c onstant c > 0 with the fol lowing pr op erty. Fix an inte g e r t > 2 , δ , ξ ∈ (0 , 1) , and D > 2 , and assume that n ∈ N satisfies n > 1 ξ D c ( t log t ) /δ . (9) L et ( X , d X ) b e a metric sp ac e and f : B n → X a D - vertic al ly faithful emb e dding. Then ther e exists a mapping φ : B t → B n with the fol lowing pr op erties. • If x, y ∈ B t ar e such that x is an anc estor of y , then φ ( x ) is an anc estor of φ ( y ) . • dist( φ ) 6 1 + ξ . • The mapping f ◦ φ : B t → X is a (1 + δ ) -vertic al ly faithful emb e dding of B t in X . Theorem 1.14 is essenti ally d ue to Matou ˇ sek [18]. Matou ˇ sek actually pro v ed this statemen t only for t = 2, since this is all that he needed in ord er to analyze forks. But, h is pro of extends in a straigh tforw ard w a y to an y t ∈ N . Since w e will use this assertion with larger t , for the sake of completeness we repro v e it, in a somewhat differen t wa y , in S ection 5. Note that Theorem 1.14 sa ys that { B n } ∞ n =1 do ha ve a lo cal rigidit y prop erty w ith resp ect to v ertically faithfully em b eddings. W e solv e the prob lem created b y the existence of configurations as those depicted in Figure 2 by studying (1 + δ )-v ertically faithful em b edd ings of B 4 , and arguing that they must conta in a large con tracted pair of p oin ts. This claim, formalized in Lemma 6.27, is prov ed in Sections 6.2, 6.3. W e b egin in Section 6.2.1 w ith studyin g ho w the metric P 2 (3-p oin t path) can b e appr o xim ately em b edded in ( B ∞ , d ε ). W e find that there are essentia lly only t wo wa ys to embed it in ( B ∞ , d ε ), as dep icted in Figure 3. W e then pr o ceed in Section 6.2.2 to study δ -forks in ( B ∞ , d ε ). Since forks are formed by “stitc hin g” tw o appr o ximate P 2 metrics along a common edge (the hand le), we can limit th e “searc h space” using the results of Section 6.2.1. W e fi nd that there are six p ossible t yp es of different appro ximate forks in ( B ∞ , d ε ), only four of wh ich (depicted in Figure 4) do n ot ha v e h ighly con tracted pron gs. Complete binary tr ees, and in particular B 4 , are comp osed of forks stitc hed together, han d le to prong. In order to study h andle-to-prong stitc hing, we in v estigate in Section 6.2.3 ho w the metric P 3 (4-p oin t path) can b e appro ximately em b edd ed in ( B ∞ , d ε ). This is again done by stud ying ho w tw o P 2 metrics can b e stitc hed together, this time b ottom edge to top ed ge. W e find that there are only three d ifferent appr oximate configurations of P 4 in ( B ∞ , d ε ). Using the mac h inery describ ed abov e, we stu dy in Section 6.3 ho w the differen t t yp es of forks can b e stitc hed together in em b edd ings of B 4 in to ( B ∞ , d ε ), r eac hing the conclusion that a large con- traction is unav oidable, and thus completing the pro of of Lemma 6.27. The pro ofs of Theorem 1.10 and Th eorem 1.12 are concluded in Section 6.4. 2 Mark o v p -con v exit y and p -con ve xit y coincide In this sec tion w e pro ve Theorem 1.3, i.e., th at for Banac h spaces p -con vexit y and Marko v p - con vexit y are the s ame pr op erties. W e first sho w that p -conv exit y implies Mark o v p -con ve xit y , and in fact it implies a str on ger inequalit y that is stated in Prop osition 2.1 b elo w. The sligh tly w eak er assertion that p -conv exit y imp lies Marko v p -con vexit y wa s first p ro v ed in [16], based on an argumen t from [5]. Ou r argumen t here is differen t and simpler. 9 It w as pro v ed in [29] that a Banac h space X is p -conv ex if and only if it admits an equiv alen t norm k · k for wh ic h there exists K > 0 su ch that f or ev ery a, b ∈ X , 2 k a k p + 2 K p k b k p 6 k a + b k p + k a − b k p . (10 ) Prop osition 2.1. L et { X t } t ∈ Z b e r andom variables taking values in a set Ω . F or every s ∈ Z let n e X t ( s ) o t ∈ Z b e r andom variables taking values in Ω , with the fol lowing pr op e rty: ∀ r 6 s 6 t, ( X r , X t ) and  X r , e X t ( s )  ha v e the same distribution . (11) Fix p > 2 and let ( X , k · k ) b e a Banach sp ac e whose norm satisfies (10) . Then for every f : Ω → X we have ∞ X k =0 X t ∈ Z E h    f ( X t ) − f  e X t ( t − 2 k )     p i 2 k p 6 (4 K ) p X t ∈ Z E  k f ( X t ) − f ( X t − 1 ) k p  . (12) Remark 2.2. Ob serve that c ondition (11) holds when { X t } t ∈ Z is a Markov chain on a state sp ac e Ω , and n e X t ( s ) o t ∈ Z is as in Definition 1.2. W e start by p ro ving a useful inequality that is a sim p le consequence of (10). Lemma 2.3. L et X b e a Banach sp ac e whose norm satisfies (10) . Then for every x, y , z , w ∈ X , k x − w k p + k x − z k p 2 p − 1 + k z − w k p 4 p − 1 K p 6 k y − w k p + k z − y k p + 2 k y − x k p . (13) Pr o of. F or ev ery x, y , z , w ∈ X , (10) implies that k x − w k p 2 p − 1 + 2 K p     y − x + w 2     p 6 k y − x k p + k y − w k p , and k z − x k p 2 p − 1 + 2 K p     y − z + x 2     p 6 k z − y k p + k y − x k p . Summing these t wo inequ alities, and applying the con v exit y of the map u 7→ k u k p , we see that k y − w k p + k z − y k p + 2 k y − x k p > k x − w k p + k z − x k p 2 p − 1 + 4 K p ·   y − x + w 2   p +   y − z + x 2   p 2 > k x − w k p + k z − x k p 2 p − 1 + 4 K p ·     z − w 4     p , implying (13). Pr o of of Pr op osition 2.1. Using L emm a 2.3 w e see that for ev ery t ∈ Z and k ∈ N , k f ( X t ) − f ( X t − 2 k ) k p + k f ( e X t ( t − 2 k − 1 )) − f ( X t − 2 k ) k p 2 p − 1 + k f ( X t ) − f ( e X t ( t − 2 k − 1 )) k p 4 p − 1 K p 6 k f ( X t − 2 k − 1 ) − f ( X t ) k p + k f ( X t − 2 k − 1 ) − f ( e X t ( t − 2 k − 1 )) k p + 2 k f ( X t − 2 k − 1 ) − f ( X t − 2 k ) k p . 10 T aking exp ectatio n, and using the assu m ption (11), w e get E  k f ( X t ) − f ( X t − 2 k ) k p  2 p − 2 + E h k f ( X t ) − f ( e X t ( t − 2 k − 1 )) k p i 4 p − 1 K p 6 2 E  k f ( X t − 2 k − 1 ) − f ( X t ) k p  + 2 E  k f ( X t − 2 k − 1 ) − f ( X t − 2 k ) k p  . Dividing by 2 ( k − 1) p +2 this b ecomes E  k f ( X t ) − f ( X t − 2 k ) k p  2 k p + E h k f ( X t ) − f ( e X t ( t − 2 k − 1 )) k p i 2 ( k +1) p K p 6 E  k f ( X t − 2 k − 1 ) − f ( X t ) k p  2 ( k − 1) p +1 + E  k f ( X t − 2 k − 1 ) − f ( X t − 2 k ) k p  2 ( k − 1) p +1 . Summing this inequalit y o v er k = 1 , . . . , m and t ∈ Z w e get m X k =1 X t ∈ Z E  k f ( X t ) − f ( X t − 2 k ) k p  2 k p + m X k =1 X t ∈ Z h E k f ( X t ) − f ( e X t ( t − 2 k − 1 )) k p i 2 ( k +1) p K p 6 m X k =1 X t ∈ Z E  k f ( X t − 2 k − 1 ) − f ( X t ) k p  2 ( k − 1) p +1 + m X k =1 X t ∈ Z E  k f ( X t − 2 k − 1 ) − f ( X t − 2 k ) k p  2 ( k − 1) p +1 = m − 1 X j =0 X s ∈ Z E [ k f ( X s ) − f ( X s − 2 j ) k p ] 2 j p . (14) It is only of inte rest to p r o ve (12) when P t ∈ Z E  k f ( X t ) − f ( X t − 1 ) k p  < ∞ . By the triangle inequalit y , this implies that for every k ∈ N w e hav e P t ∈ Z E  k f ( X t ) − f ( X t − 2 k ) k p  < ∞ . W e ma y therefore cancel terms in (14), arriving at the follo wing inequalit y: m X k =1 X t ∈ Z E h k f ( X t ) − f ( e X t ( t − 2 k − 1 )) k p i 2 ( k +1) p K p 6 X t ∈ Z E [ k f ( X t ) − f ( X t − 1 ) k p ] − X t ∈ Z E [ k f ( X t ) − f ( X t − 2 m ) k p ] 2 mp 6 X t ∈ Z E [ k f ( X t ) − f ( X t − 1 ) k p ] . Equiv alen tly , m − 1 X k =0 X t ∈ Z E h k f ( X t ) − f ( e X t ( t − 2 k )) k p i 2 k p 6 (4 K ) p X t ∈ Z E [ k f ( X t ) − f ( X t − 1 ) k p ] . Prop osition 2.1 no w follo ws by letting m → ∞ . W e next pro v e the m ore in teresting direction of the equiv alence of p -con v exit y and Marko v p -con vexit y: a Mark ov p -con v ex Banac h space is also p -conv ex. 11 Theorem 2.4. L et ( X, k · k ) b e a Banach sp ac e which i s Markov p -c onvex with c onstant Π . Then for every ε ∈ (0 , 1) ther e exists a norm | | | · | | | on X such that for al l x, y ∈ X , (1 − ε ) k x k 6 | | | x | | | 6 k x k , and             x + y 2             p 6 | | | x | | | p + | | | y | | | p 2 − 1 − (1 − ε ) p 4Π p ( p + 1) ·             x − y 2             p . Thus the norm | | | · | | | satisfies (10) with c onstant K = O  Π ε 1 /p  . Pr o of. The f act that X is Mark ov p -con v ex with constan t Π implies that for ev ery Marko v c hain { X t } t ∈ Z with v alues in X , and for eve ry m ∈ N , we ha v e m X k =0 2 m X t =1 E h   X t − e X t ( t − 2 k )   p i 2 k p 6 Π p 2 m X t =1 E [ k X t − X t − 1 k p ] . (15) F or x ∈ X we sh all sa y that a Mark o v chain { X t } 2 m t = −∞ is an m -adm iss ible r epresen tation of x if X t = 0 for t 6 0 and E [ X t ] = tx for t ∈ { 1 , . . . , 2 m } . Fix ε ∈ (0 , 1), and denote η = 1 − (1 − ε ) p . F or every m ∈ N d efine | | | x | | | m = inf        1 2 m 2 m X t =1 E [ k X t − X t − 1 k p ] − η Π p · 1 2 m m X k =0 2 m X t =1 E h   X t − e X t ( t − 2 k )   p i 2 k p   1 /p      , (16) where the infim um in (16) is tak en o ver all m -admissible repr esen tations of x . Observe that an m -admissible repr esentati on of x alw a ys exists, sin ce w e can defin e X t = 0 for t 6 0 and X t = tx for t ∈ { 1 , . . . , 2 m } . This example shows that | | | x | | | m 6 k x k . On the other hand, if { X t } 2 m t = −∞ is an m -admissible represen tation of x then 2 m X t =1 E [ k X t − X t − 1 k p ] − η Π p m X k =0 2 m X t =1 E h   X t − e X t ( t − 2 k )   p i 2 k p (15) > (1 − η ) 2 m X t =1 E [ k X t − X t − 1 k p ] > (1 − ε ) p 2 m X t =1 k E [ X t ] − E [ X t − 1 ] k p = (1 − ε ) p 2 m X t =1 k tx − ( t − 1) x k p = 2 m (1 − ε ) p k x k p , (17 ) where in the first inequalit y of (17) we used th e con v exit y of the function z 7→ k z k p . In conclusion, w e see that for all x ∈ X , (1 − ε ) k x k 6 | | | x | | | m 6 k x k . (18) No w tak e x, y ∈ X and fix δ ∈ (0 , 1). Let { X t } 2 m t = −∞ b e an adm issible represen tation on x and { Y t } 2 m t = −∞ b e an admissib le r epresent ation of y which is sto chasti cally indep endent of { X t } 2 m t = −∞ , suc h that 2 m X t =1 E [ k X t − X t − 1 k p ] − η Π p m X k =0 2 m X t =1 E h   X t − e X t ( t − 2 k )   p i 2 k p 6 2 m ( | | | x | | | p m + δ ) , (19) 12 and 2 m X t =1 E [ k Y t − Y t − 1 k p ] − η Π p m X k =0 2 m X t =1 E h   Y t − e Y t ( t − 2 k )   p i 2 k p 6 2 m ( | | | y | | | p m + δ ) . (20) Define a Mark o v chain { Z t } 2 m +1 t = −∞ ⊆ X as follo ws. F or t 6 − 2 m set Z t = 0. With probabilit y 1 2 let ( Z − 2 m +1 , Z − 2 m +2 , . . . , Z 2 m +1 ) equal  0 , . . . , 0 | {z } 2 m times , X 1 , X 2 , . . . , X 2 m , X 2 m + Y 1 , X 2 m + Y 2 , . . . , X 2 m + Y 2 m  , and with probabilit y 1 2 let ( Z − 2 m +1 , Z − 2 m , . . . , Z 2 m +1 ) equal  0 , . . . , 0 | {z } 2 m times , Y 1 , Y 2 , . . . , Y 2 m , X 1 + Y 2 m , X 2 + Y 2 m , . . . , X 2 m + Y 2 m  . Hence, Z t = 0 for t 6 0, for t ∈ { 1 , . . . , 2 m } we ha v e E [ Z t ] = E [ X t ]+ E [ Y t ] 2 = t · x + y 2 , and for t ∈ { 2 m + 1 , . . . , 2 m +1 } we ha v e E [ Z t ] = E [ X 2 m + Y t − 2 m ] + E [ X t − 2 m + Y 2 m ] 2 = 2 m x + ( t − 2 m ) y + ( t − 2 m ) x + 2 m y 2 = t · x + y 2 . Th us { Z t } 2 m +1 t = −∞ is an ( m + 1)-admissible r epresent ation of x + y 2 . The d efinition (16) imp lies that 2 m +1             x + y 2             p m +1 6 2 m +1 X t =1 E [ k Z t − Z t − 1 k p ] − η Π p m +1 X k =0 2 m +1 X t =1 E h   Z t − e Z t ( t − 2 k )   p i 2 k p . (21) Note that b y d efi nition, 2 m +1 X t =1 E [ k Z t − Z t − 1 k p ] = 2 m X t =1 E [ k X t − X t − 1 k p ] + 2 m X t =1 E [ k Y t − Y t − 1 k p ] . (22) Moreo ve r, m +1 X k =0 2 m +1 X t =1 E h   Z t − e Z t ( t − 2 k )   p i 2 k p = 1 2 ( m +1) p 2 m +1 X t =1 E h   Z t − e Z t ( t − 2 m +1 )   p i + m X k =0 2 m +1 X t =1 E h   Z t − e Z t ( t − 2 k )   p i 2 k p . (23 ) W e b ound eac h of the terms in ( 23) sep arately . Note that b y c onstruction w e ha ve for every t ∈ { 1 , . . . , 2 m } , Z t − e Z t  t − 2 m +1  = Z t − e Z t  1 − 2 m +1  =            X t − Y t with pr obabilit y 1 / 4 , Y t − X t with pr obabilit y 1 / 4 , X t − e X t (1) with p robabilit y 1 / 4 , Y t − e Y t (1) with probabilit y 1 / 4 . 13 Th us, the first term in the righ t h and side of (23) can b e b ounded from b elow as follo ws: 1 2 ( m +1) p 2 m +1 X t =1 E h   Z t − e Z t ( t − 2 m +1 )   p i > 1 2 ( m +1) p +1 2 m X t =1 E [ k X t − Y t k p ] > 1 2 ( m +1) p +1 2 m X t =1 k E [ X t ] − E [ Y t ] k p = k x − y k p 2 ( m +1) p +1 2 m X t =1 t p > 2 m k x − y k p 2 p +1 ( p + 1) . (24 ) W e no w pr o ceed to b ound from b elo w the second term in the righ t hand side of (23) . Note first that for ev ery k ∈ { 0 , . . . , m } and ev ery t ∈ { 2 m + 1 , . . . , 2 m +1 } we ha v e Z t − e Z t ( t − 2 k ) =    ( X 2 m + Y t − 2 m ) −  e X 2 m ( t − 2 k ) + e Y t − 2 m ( t − 2 m − 2 k )  with pr obabilit y 1 / 2 , ( Y 2 m + X t − 2 m ) −  e Y 2 m ( t − 2 k ) + e X t − 2 m ( t − 2 m − 2 k )  with pr obabilit y 1 / 2 . By Jensen’s inequality , if U, V are X -v alued indep endent r andom v ariables with E [ V ] = 0, then E [ k U + V k p ] > E [ k U + E [ V ] k p ] = E [ k U k p ]. T h us, since { X t } 2 m t = −∞ and { Y t } 2 m t = −∞ are in dep end en t, E h   Y t − 2 m − e Y t − 2 m ( t − 2 m − 2 k ) + X 2 m − e X 2 m ( t − 2 k )   p i > E h   Y t − 2 m − e Y t − 2 m ( t − 2 m − 2 k )   p i , and E h   X t − 2 m − e X t − 2 m ( t − 2 m − 2 k ) + Y 2 m − e Y 2 m ( t − 2 k )   p i > E h   X t − 2 m − e X t − 2 m ( t − 2 m − 2 k )   p i . It follo w s that for ev ery k ∈ { 0 , . . . , m } and ev ery t ∈ { 2 m + 1 , . . . , 2 m +1 } we ha v e E h   Z t − e Z t ( t − 2 k )   p i > 1 2 E h   X t − 2 m − e X t − 2 m ( t − 2 m − 2 k )   p i + 1 2 E h   Y t − 2 m − e Y t − 2 m ( t − 2 m − 2 k )   p i . (25 ) Hence, m X k =0 2 m +1 X t =1 E h   Z t − e Z t ( t − 2 k )   p i 2 k p (25) > m X k =0 2 m X t =1 1 2 E h   X t − e X t ( t − 2 k )   p i + 1 2 E h   Y t − e Y t ( t − 2 k )   p i 2 k p + m X k =0 2 m +1 X t =2 m +1 1 2 h E   X t − 2 m − e X t − 2 m ( t − 2 m − 2 k )   p i + 1 2 E h   Y t − 2 m − e Y t − 2 m ( t − 2 m − 2 k )   p i 2 k p = m X k =0 2 m X t =1 E h   X t − e X t ( t − 2 k )   p i 2 k p + m X k =0 2 m X t =1 E h   Y t − e Y t ( t − 2 k )   p i 2 k p . (26) Com bining (19), (20), (21), (22), (23), (24) and (26), and letting δ tend to 0, we see th at 2 m +1             x + y 2             p m +1 6 2 m | | | x | | | p m + 2 m | | | y | | | p m − η Π p · 2 m k x − y k p 2 p +1 ( p + 1) , 14 or,             x + y 2             p m +1 6 | | | x | | | p m + | | | y | | | p m 2 − η 4Π p ( p + 1) ·     x − y 2     p . (27) Define f or w ∈ X , | | | w | | | = lim sup m →∞ | | | w | | | m . Then a com bination of (18) and (27) yields that (1 − ε ) k x k 6 | | | x | | | 6 k x k , and             x + y 2             p 6 | | | x | | | p + | | | y | | | p 2 − η 4Π p ( p + 1) ·     x − y 2     p 6 | | | x | | | p + | | | y | | | p 2 − η 4Π p ( p + 1) ·             x − y 2             p . (28 ) Note that (28) imp lies that the set { x ∈ X : | | | x | | | 6 1 } is con v ex, so that | | | · | | | is a norm on X . This conclud es the pro of of Theorem 2.4. 3 A doubling space whic h is not Mark o v p -con v ex for an y p ∈ (0 , ∞ ) G 1 G 0 G 2 G 3 r r r r b Consider the Laakso graphs [12], { G i } ∞ i =0 , wh ich are defined as follo ws. G 0 is th e graph consisting of one edge of unit length. T o constru ct G i , tak e six copies of G i − 1 and scale their metric by a factor of 1 4 . W e glue four of them cyclicly by iden tifying pairs of end p oints, and attac h at tw o opp osite gluing p oints th e remaining t wo copies. Note th at eac h edge of G i has length 4 − i ; we denoted the resulting shortest path metric on G i b y d G i . As sho wn in [13, T hm. 2.3], the doubling constan t of metric space ( G i , d G i ) is at most 6. W e d ir ect G m as follo w s . Define the ro ot of G m to b e (an arbitrarily c h osen) one of the tw o vertices havi ng only one adjacent edge. I n the fi gure this could b e the leftmost v ertex r . Note that in no edge the t w o endp oin ts are at the same distance from th e ro ot. The edges of G m are then dir ected from the endp oin t closer to the ro ot to the en d p oint further aw a y from the ro ot. The r esulting d irected graph is acyclic. W e no w define { X t } 4 m t =0 to b e the standard random w alk on the directed graph G m , starting from the ro ot. This random wa lk is extended to t ∈ Z b y stipu lating that X t = X 0 for t < 0, and X t = X 4 m for t > 4 m . Prop osition 3.1. F or the r andom walk define d ab ove, 2 m X k =0 X t ∈ Z E h d G m  X t , e X t ( t − 2 k )  p i 2 k p & m 8 p X t ∈ Z E [ d G m ( X t , X t − 1 ) p ] . (29) 15 Pr o of. F or ev ery t ∈ Z w e ha ve , E [ d G m ( X t , X t − 1 ) p ] =  4 − mp t ∈ { 0 , . . . 4 m − 1 } , 0 otherwise . Hence, X t ∈ Z E [ d G m ( X t , X t − 1 ) p ] = 4 − m ( p − 1) . (30) Fix k ∈ { 0 , . . . , 2 m − 2 } and write h = ⌈ k / 2 ⌉ . Vie w G m as b eing built fr om A = G m − h , w here eac h edge of A has b een replaced by a copy of G h . Note that for every i ∈ { 0 , . . . , 4 m − h − 1 + 1 } , at time t = (4 i + 1)4 h the wa lk X t is at a v ertex of G m whic h h as t w o outgoing edges, corresp onding to distinct copies of G h . T o see th is it suffices to sho w that all vertic es of G m − h that are exactly (4 i + 1) edges aw a y from the ro ot, h a ve out-degree 2. This fact is true since G m − h is obtained from G m − h − 1 b y replacing eac h edge b y a copy of G 1 , and eac h such cop y of G 1 con trib utes one vertex of out-degree 2, corresp onding to the v ertex lab eled b in the figure describing G 1 . Consider the s et of times T k def = { 0 , . . . , 4 m − 1 } \   4 m − h − 1 +1 [ i =0  (4 i + 1)4 h + 4 h − 2 , (4 i + 1)4 h + 2 · 4 h − 2    . F or t ∈ T k find i ∈ { 0 , . . . , 4 m − h − 1 + 1 } su c h that t ∈  (4 i + 1)4 h + 4 h − 2 , (4 i + 1)4 h + 2 · 4 h − 2  . Since, by th e defin ition of h , we h a ve t − 2 k ∈  (4 i + 1)4 h − 4 h , (4 i + 1)4 h  , the wa lks { X s } s ∈ Z and { e X s ( t − 2 k ) } s ∈ Z started ev olving indep en d en tly at some ve rtex lying in a cop y of G h preceding a v ertex v of G m whic h has t wo outgoing edges, corresp ondin g to distinct copies of G h . Thus, with probab ility at least 1 2 , the walks X t and e X t ( t − 2 k ) lie on tw o distinct copies of G h in G m , immediately follo wing the v ertex v , and at distance at least 4 h − 2 · 4 − m and at most 2 · 4 h − 2 · 4 − m from v . Hence, with probabilit y at least 1 2 w e h av e d G m  X t , e X t ( t − 2 k )  > 2 · 4 h − 2 · 4 − m = 2 2 h − 3 − 2 m , and therefore, E h d ( X t , e X t ( t − 2 k )) p i 2 k p > 1 2 2 (2 h − 3 − 2 m ) p 2 k p > 2 − (2 m +3) p − 1 . W e deduce that for all k ∈ { 0 , . . . , 2 m − 2 } , X t ∈ Z E h d ( X t , e X t ( t − 2 k )) p i 2 k p > X t ∈ T k E h d ( X t , e X t ( t − 2 k )) p i 2 k p > | T k | · 2 − (2 m +3) p − 1 & 4 h − 2 · 4 m − h − 1 · 2 − (2 m +3) p − 1 & 1 8 p 4 − m ( p − 1) . (31 ) A com bination of (30) and (31) imp lies (29) . Pr o of of The or em 1.5. As explained in [12, 13], b y passing to an app ropriate Gromo v-Hausdorff limit, there exists a doubling met ric space ( X, d X ) that con tains an isometric cop y of all the Laakso graphs { G m } ∞ m =0 . Prop osition 3.1 therefore implies that X is not Mark ov p -conv ex for any p ∈ (0 , ∞ ). 16 Pr o of of The or em 1.6. Let ( X , d X ) b e a Marko v p -conv ex metric space, i.e, Π p ( X ) < ∞ . Assume that f : G m → X satisfies x, y ∈ G m = ⇒ 1 A d G m ( x, y ) 6 d X ( f ( x ) , f ( y )) 6 B d G m ( x, y ) . (32) Let { X t } t ∈ Z b e th e random w alk from Pr op osition 3.1. Then m 8 p A p X t ∈ Z E [ d G m ( X t , X t − 1 ) p ] (29) . 1 A p 2 m X k =0 X t ∈ Z E h d G m  X t , e X t ( t − 2 k )  p i 2 k p (32) 6 2 m X k =0 X t ∈ Z E h d X  f ( X t ) , f ( e X t ( t − 2 k ))  p i 2 k p (3) 6 Π p ( X ) p X t ∈ Z E [ d X ( f ( X t ) , f ( X t − 1 )) p ] (32) 6 Π p ( X ) p B p X t ∈ Z E [ d G m ( X t , X t − 1 ) p ] . Th us AB & m 1 /p & (log | G m | ) 1 /p . 4 Lipsc hitz quotien ts Sa y that a metric space ( Y , d Y ) is a D -Lipsc hitz quotient of a m etric space ( X , d X ) if there exist a, b > 0 with ab 6 D and a mapping f : X → Y s u c h that for all x ∈ X and r > 0, B Y  f ( x ) , r a  ⊆ f ( B X ( x, r )) ⊆ B Y ( f ( x ) , br ) . (33) Observe that the last inclusion in (33 ) is to equiv alen t to the fact that f is b -Lipsc h itz. The follo wing prop osition implies Th eorem 1.4. Prop osition 4.1. If ( Y , D Y ) is a D -Lipschitz quotient of ( X , d X ) then Π p ( Y ) 6 D · Π p ( X ) . Pr o of. Fix f : X → Y s atisfying (33). Also, fix a Mark ov chain { X t } t ∈ Z on a state space Ω , and a mapping g : Ω → Y . Fix m ∈ Z and let Ω ∗ b e the set of finite sequences of elemen ts of Ω starting at time m , i.e., the s et of sequences of the form ( ω i ) t i = m ∈ Ω t − m +1 for all t > m . It will b e con venien t to consider the Marko v c h ain { X ∗ t } ∞ t = m on Ω ∗ whic h is giv en by: Pr [ X ∗ t = ( ω m , ω m +1 , . . . , ω t )] = Pr [ X m = ω m , X m +1 = ω m +1 , . . . , X t = ω t ] . Also, define g ∗ : Ω ∗ → Y by g ∗ ( ω 1 , . . . , ω t ) = g ( ω t ). By d efinition, { g ∗ ( X ∗ t ) } ∞ t = m and { g ( X t ) } ∞ t = m are identical ly distributed. W e next define a mapping h ∗ : Ω ∗ → X su c h that f ◦ h ∗ = g ∗ and for all ( ω m , . . . , ω t ) ∈ Ω ∗ , d X ( h ∗ ( ω m , . . . , ω t − 1 ) , h ∗ ( ω m , . . . , ω t )) 6 ad Y ( g ( ω t − 1 ) , g ( ω t )) . (34) 17 F or ω ∗ ∈ Ω ∗ , w e will d efine h ∗ ( ω ∗ ) by induction on the length of ω ∗ . I f ω ∗ = ( ω m ), th en we fix h ∗ ( ω ∗ ) to b e an arbitrary element in f − 1 ( g ( ω m )). Assum e that ω ∗ = ( ω m , . . . , ω t − 1 , ω t ) and that h ∗ ( ω m , . . . , ω t − 1 ) h as b een defi n ed. Set x = f ( h ∗ ( ω m , . . . , ω t − 1 )) = g ∗ ( ω m , . . . , ω t − 1 ) = g ( ω t − 1 ) and r = ad Y ( g ( ω t − 1 ) , g ( ω t )). Since g ( ω t ) ∈ B Y ( x, r/a ), it f ollo ws fr om (33) there exists y ∈ X suc h that f ( y ) = g ( ω t ), and d X ( x, y ) 6 r . W e then define h ∗ (( ω m , . . . , ω t − 1 , ω t )) def = y . W rite X ∗ t = X ∗ m for t 6 m . By the Mark o v p -con v exit y of ( X , d X ), we h a ve ∞ X k =0 X t ∈ Z E h d X  h ∗ ( X ∗ t ) , h ∗ ( e X ∗ t ( t − 2 k ))  p i 2 k p 6 Π p ( X ) p X t ∈ Z E  d X ( h ∗ ( X ∗ t ) , h ∗ ( X ∗ t − 1 )) p  . (35) By (34) we ha v e for ev ery t > m + 1, d X ( h ∗ ( X ∗ t ) , h ∗ ( X ∗ t − 1 )) 6 ad Y ( g ( X t ) , g ( X t − 1 )) , while for t 6 m w e h a ve d X ( h ∗ ( X ∗ t ) , h ∗ ( X ∗ t − 1 )) = 0. Thus, X t ∈ Z E  d X ( h ∗ ( X ∗ t ) , h ∗ ( X ∗ t − 1 )) p  6 a p X t ∈ Z E [ d Y ( g ( X t ) , g ( X t − 1 )) p ] . (36) A t the same time, u sing the fact that f is b -Lip s c h itz and f ◦ h ∗ = g ∗ , we see that if t > m + 2 k , d X  h ∗ ( X ∗ t ) , h ∗ ( e X ∗ t ( t − 2 k ))  > 1 b d Y  f ( h ∗ ( X ∗ t )) , f ( h ∗ ( e X ∗ t ( t − 2 k )))  = 1 b d Y  g ∗ ( X ∗ t ) , g ∗ ( e X ∗ t ( t − 2 k ))  = 1 b d Y  g ( X t ) , g ( e .X t ( t − 2 k ))  Th us, ∞ X k =0 X t ∈ Z E h d X  h ∗ ( X ∗ t ) , h ∗ ( e X ∗ t ( t − 2 k ))  p i 2 k p > 1 b p ∞ X k =0 ∞ X t = m +2 k E h d Y  g ( X t ) , g ( e X t ( t − 2 k ))  p i 2 k p . (3 7) By combining (36) and (37) with (35), and letting m tend to −∞ , we get th e in equ alit y: ∞ X k =0 X t ∈ Z E h d Y  g ( X t ) , g ( e X t ( t − 2 k ))  p i 2 k p 6 ( ab Π p ( X )) p X t ∈ Z E [ d Y ( g ( X t ) , g ( X t − 1 )) p ] . Since th is inequalit y h olds f or every Marko v c hain { X t } t ∈ Z and ev ery g : Ω → Y , and since ab 6 D , w e h a ve prov ed that Π p ( Y ) 6 D Π p ( X ), as required. 5 A d ic hotom y theorem for v ertically faithful em b eddings of trees In this s ection we pro v e T heorem 1.14. The pr o of naturally breaks into tw o parts. The fi rst is the follo wing BD Ramsey prop ert y of paths (whic h can b e found n on-quan titativ ely in [19], wh ere also the BD Ramsey terminology is explained). A m apping φ : M → N is called a r esc ale d isometry if dist( φ ) = 1, or equiv alently there exists λ > 0 such that d N ( φ ( x ) , φ ( y )) = λd M ( x, y ) for all x, y ∈ M . F or n ∈ N let P n denote th e n -path, i.e., th e set { 0 , . . . , n } equipp ed with the metric inherited from the real line. 18 Prop osition 5.1. Fix δ ∈ (0 , 1) , D > 2 and t, n ∈ N satisfying n > D (4 t log t ) /δ . If f : P n → X satisfies dist( f ) 6 D then ther e exists a r e sc ale d isometry φ : P t → P n such that d ist( f ◦ φ ) 6 1 + δ . Giv en a metric space ( X , d X ) and a nonconstant m apping f : P n → X , defin e T ( X, f ) def = d X ( f (0) , f ( n ) n max i ∈{ 1 ,...,n } d X ( f ( i − 1) , f ( i )) = d X ( f (0) , f ( n )) n k f k Lip . If f is a constan t mapping (equiv alent ly m ax i ∈{ 1 ,...,n } d X ( f ( i − 1) , f ( i )) = 0) then we set T ( X , f ) = 0. Note that b y the triangle inequalit y we alwa ys ha v e T ( X , f ) 6 1. Lemma 5.2. F or every m, n ∈ N and f : P mn → X , ther e exist r esc ale d isometries φ ( n ) : P n → P mn and φ ( m ) : P m → P mn , such that T ( X, f ) 6 T  X, f ◦ φ ( m )  · T  X, f ◦ φ ( n )  . Pr o of. Fix f : P mn → X and define φ ( m ) : P m → P mn b y φ ( m ) ( i ) = in . Then , d X ( f (0) , f ( mn )) 6 T  X, f ◦ φ ( m )  m max i ∈{ 1 ,...,m } d X ( f (( i − 1) n ) , f ( in )) . (38) Similarly , for ev er y i ∈ { 1 , . . . , m } define φ ( n ) i : P n → P mn b y φ ( n ) i ( j ) = ( i − 1) n + j . Then d X ( f (( i − 1) n ) , f ( in )) 6 T  X, f ◦ φ ( n ) i  n max j ∈{ 1 ,...,n } d X ( f (( i − 1) n + j − 1) , f (( i − 1) n + j )) . (39) Letting i ∈ { 1 , . . . , m } b e su c h that T  X, f ◦ φ ( n ) i  is maximal, and φ ( n ) = φ ( n ) i , we conclud e that d X ( f (0) , f ( mn )) (38) ∧ (39) 6 T  X, f ◦ φ ( m )  T  X, f ◦ φ ( n )  mn max i ∈{ 1 ,...,mn } d X ( f ( i − 1) , f ( i )) . Lemma 5.3. F or every f : P m → X we have d ist( f ) > 1 /T ( X, f ) . Pr o of. Assumin g a | i − j | 6 d X ( f ( i ) , f ( j )) 6 b | i − j | for all i, j ∈ P m , the claim is bT ( X, f ) > a . Indeed, am 6 d X ( f (0) , f ( m )) 6 T ( X , f ) m m ax i = ∈{ 1 ,...,m } d X ( f ( i − 1 ) , f ( i )) 6 T ( X , f ) bm . Lemma 5.4. Fix f : P m → X . If 0 < ε < 1 /m and T ( X, f ) > 1 − ε , then dist( f ) 6 1 / (1 − mε ) . Pr o of. Denote b = max i ∈{ 1 ,...,n } d X ( f ( i ) , f ( i − 1)) > 0. F or ev ery 0 6 i < j 6 m we ha ve d X ( f ( i ) , f ( j )) 6 P j ℓ = i +1 d X ( f ( ℓ − 1) , f ( ℓ )) 6 b | j − i | , and (1 − ε ) mb 6 T ( X, f ) mb = d X ( f (0) , f ( m )) 6 d X ( f (0) , f ( i )) + d X ( f ( i ) , f ( j )) + d X ( f ( j ) , f ( m )) 6 d X ( f ( i ) , f ( j )) + b ( m + i − j ) . Th us d X ( f ( i ) , f ( j )) > b ( j − i − m ε ) > (1 − mε ) b | j − i | . Pr o of of Pr op osition 5.1. Set k = ⌊ log t n ⌋ and d enote b y I the identit y mapp ing fr om P t k to P n . By Lemma 5.3 we hav e T ( X, f ◦ I ) > 1 /D . An iterativ e application of Lemm a 5.2 imp lies that there exists a rescaled isometry φ : P t → P t k suc h that T ( X, f ◦ I ◦ φ ) > D − 1 /k > e − 2 log D / log t n > e − δ/ (2 t ) > 1 − δ 2 t . By Lemm a 5.4 we therefore ha v e dist( f ◦ I ◦ φ ) 6 1 / (1 − δ / 2) 6 1 + δ . 19 The second part of the p ro of of Theorem 1.14 uses the follo win g com binatorial lemma due to Matou ˇ sek [18]. Denote by T k ,m the complete ro oted tree of heigh t m , in w hic h ev er y non-leaf vertex has k children. F or a ro oted tree T , denote by SP( T ) the set of all u n ordered pairs { x, y } of distinct v ertices of T su c h that x is an ancestor of y . Lemma 5.5 ([18, Lem. 5]) . L et m, r, k ∈ N satisfy k > r ( m +1) 2 . Supp ose that e ach of the p airs fr om SP( T k ,m ) is c olor e d by one of r c olors. Then ther e exists a c opy T ′ of B m in this T k ,m such that the c olor of any p air { x, y } ∈ SP( T ′ ) only dep ends on the levels of x and y . Pr o of of L emma 1.14. Let f : B n → X b e a D -v ertically faithful emb edding, i.e., for some λ > 0 it satisfies λd B n ( x, y ) 6 d X ( f ( x ) , f ( y )) 6 D λd B n ( x, y ) (40) whenev er x, y ∈ B n are su c h that x is an ancestor of y . Let k , ℓ ∈ N b e auxiliary p arameters to b e determin ed later, and defin e m = ⌊ n/ ( k ℓ ) ⌋ . W e first construct a mapp ing g : T 2 k ,m → B n in a top-d own manner as follo ws. If r is the ro ot of T 2 k ,m then g ( r ) is d efined to b e the ro ot of B n . Ha ving defined g ( u ), let v 1 , . . . , v 2 k ∈ T 2 k ,m b e the c h ildren of u , and let w 1 , . . . , w 2 k ∈ B n b e the descendants of g ( u ) at depth k b elo w g ( u ). F or eac h i ∈ { 1 , . . . , 2 k } let g ( v i ) b e an arb itrary descendant of w i at depth h ( g ( u )) + ℓk . Note that for this construction to b e p ossible we need to ha v e mℓ k 6 n , whic h is ensured b y our c hoice of m . By construction, if x, y ∈ T 2 k ,m and x is an ancestor of y , then g ( x ) is an ancestor of g ( y ) and d B n ( g ( x ) , g ( y )) = ℓ k d T 2 k ,m ( x, y ). Al so, if x, y ∈ T 2 k ,m and lca ( x, y ) = u , then we hav e h ( lca ( g ( x ) , g ( y ))) ∈ { h ( g ( u )) , h ( g ( u )) + 1 , . . . , h ( g ( u )) + k − 1 } . This implies that (( ℓ − 1) k + 1) d T 2 k ,m ( x, y ) 6 d B n ( x, y ) 6 ℓk d T 2 k ,m ( x, y ) . Th us, assuming ℓ > 2, w e hav e d ist( g ) 6 1 + 2 /ℓ . Moreo v er, denoting F = f ◦ g and us ing (40), we see that if x, y ∈ T 2 k ,m are such that x is an ancestor of y then k ℓλd T 2 k ,m ( x, y ) 6 d X ( F ( x ) , F ( y )) 6 D ℓk λd T 2 k ,m ( x, y ) . (41) Color every pair { x, y } ∈ SP( T 2 k ,m ) with the color χ ( { x, y } ) def = $ log 1+ δ/ 4 d X ( F ( x ) , F ( y )) k ℓλd T 2 k ,m ( x, y ) !% ∈ { 1 , . . . , r } , where r = ⌈ log 1+ δ/ 4 D ⌉ . Assu ming that 2 k > r ( m +1) 2 , (42) b y Lemma 5.5 there exists a cop y T ′ of B m in T 2 k ,m suc h that the colors of pairs { x, y } ∈ SP( T ′ ) only dep end on the lev els of x and y . Let P b e a ro ot-leaf path in T ′ (isometric to P m ). The mappin g F | P : P → X has distortion at most D b y (41). Assu ming m > D 16( t log t ) /δ , (43) b y Pr op osition 5.1 th er e are { x i } t i =0 ⊆ P s u c h that f or some a, b ∈ N with a, a + tb ∈ [0 , m ], for all i we h a ve h ( x i ) = a + ib , and for some θ > 0, for all i, j ∈ { 0 , . . . , t } , θ b | i − j | 6 d X ( F ( x i ) , F ( x j )) 6  1 + δ 4  θ b | i − j | . (44) 20 Define a rescaled isometry ϕ : B t → T ′ in a top-do wn manner as follo ws: ϕ ( r ) = x 0 , a nd ha ving defi n ed ϕ ( u ) ∈ T ′ , if v , w are the c h ildren of u in B t and v ′ , w ′ are the c hildr en of ϕ ( u ) in T ′ , the vertice s ϕ ( v ) , ϕ ( w ) are c h osen as arbitrary d escendan ts in T ′ of v ′ , w ′ (resp ectiv ely) at depth h ( ϕ ( u )) + b . Consider the mappin g G : B t → X giv en by G = F ◦ ϕ = f ◦ g ◦ ϕ . T ake x, y ∈ B t suc h that x is an ancestor of y . W rite h ( x ) = i and h ( y ) = j . Thus h ( ϕ ( x )) = a + ib and h ( ϕ ( y )) = a + j b . It follo ws that { ϕ ( x ) , ϕ ( y ) } is colo red b y the same color as { x i , x j } , i.e.,  log 1+ δ 4  d X ( G ( x ) , G ( y )) k ℓλbd B t ( x, y )  = χ ( { ϕ ( x ) , ϕ ( y ) } ) = χ ( { x i , x j } ) =  log 1+ δ 4  d X ( F ( x i ) , F ( y j )) k ℓλbd B t ( x, y )  . Consequent ly , using (44 ) w e d educe that θ b 1 + δ / 4 d B t ( x, y ) 6 d X ( G ( x ) , G ( y )) 6  1 + δ 4  2 θ bd B t ( x, y ) . Th us G is a (1 + δ/ 4) 3 6 1 + δ v ertically faithful embed d ing of B t in to X . It remains to determine the v alues of the auxiliary parameters ℓ, k , whic h will lead to the desired restriction on n giv en in (9). First of all, we wan t to h av e dist( g ◦ ϕ ) 6 1 + ξ . Since ϕ is a rescaled isometry and (for ℓ > 2) d ist( g ) 6 1 + 2 /ℓ , w e c ho ose ℓ = ⌈ 2 /ξ ⌉ > 2. W e will c h o ose k so th at 4 k 6 nξ , so that n/ ( k ℓ ) > 1. Since m = ⌊ n/ ( k ℓ ) ⌋ , we ha v e m + 1 6 nξ /k and m > nξ / (4 k ). Recall that r = ⌈ log 1+ δ/ 4 D ⌉ 6 2 log 1+ δ/ 4 D 6 16 D /δ . Hence the r equiremen t (42) will b e satisfied if 2 k 3 >  16 D δ  n 2 ξ 2 , (45) and the requiremen t (43) will b e satisfied if nξ 4 k > D 16( t log t ) /δ . (46) There exists an inte ger k satisfying b oth (45) and (46) pro vided that  n 2 ξ 2 log 2  16 D δ  1 / 3 + 1 6 nξ 4 D 16( t log t ) /δ , whic h h olds true pro vid ed the constan t c in (9) is large enough. 6 T ree metrics do not ha v e the dic hotom y prop ert y This section is dev oted to the pr o ofs of Theorem 1.10 and T heorem 1.12. These pro ofs were outlined in Section 1.1.1 , and we will use the notation int ro du ced there. 6.1 Horizon tally contracted t rees W e start with the follo wing lemma w hic h supp lies cond itions on { ε n } ∞ n =0 ensuring that th e H -tree ( B ∞ , d ε ) is a metric space. Lemma 6.1 . Assume that { ε n } ∞ n =0 ⊆ (0 , 1] is non-incr e asing and { nε n } ∞ n =0 is non-de cr e asing. Then d ε is a metric on B ∞ 21 Pr o of. T ake x, y , z ∈ B ∞ and without loss of generalit y assu me that h ( x ) 6 h ( y ). W e distinguish b et we en the cases h ( z ) > h ( y ), h ( x ) 6 h ( z ) 6 h ( y ) and h ( z ) < h ( x ). If h ( z ) > h ( y ) then d ε ( x, z ) + d ε ( z , y ) − d ε ( x, y ) = 2[ h ( z ) − h ( y )] + 2 ε h ( x ) · [ h ( lca ( x, y )) − h ( lca ( x, z ))] + 2 ε h ( y ) · [ h ( y ) − h ( lca ( z , y ))] > 2 ε h ( x ) · [ h ( lca ( x, y )) − h ( lca ( x, z ))] + 2 ε h ( y ) · [ h ( y ) − h ( lca ( z , y ))] . (47) T o show that (47) is non-n egativ e observe that this is ob vious if h ( lca ( x, y )) > h ( lca ( x, z )). So assume that h ( lca ( x, y )) < h ( lca ( x, z )). In this case necessarily h ( lca ( z , y )) = h ( lca ( x, y )), so we can b ound (47) from b elo w as follo w s 2 ε h ( x ) · [ h ( lca ( x, y )) − h ( lca ( x, z ))] + 2 ε h ( y ) · [ h ( y ) − h ( lca ( z , y ))] > 2 ε h ( x ) · [ h ( lca ( x, y )) − h ( x )] + 2 h ( x ) h ( y ) ε h ( x ) · [ h ( y ) − h ( lca ( x, y ))] = 2 ε h ( x ) · h ( lca ( x, y ))  1 − h ( x ) h ( y )  > 0 . If h ( z ) < h ( x ) then d ε ( x, z ) + d ε ( z , y ) = h ( x ) + h ( y ) − 2 h ( z ) + 2 ε h ( z ) · [2 h ( z ) − h ( lca ( x, z )) − h ( lca ( z , y ))] > h ( y ) − h ( x ) + 2 ε h ( x ) · [ h ( x ) − h ( z )] + 2 ε h ( x ) · [2 h ( z ) − h ( lca ( x, z )) − h ( lca ( y , z ))] = h ( y ) − h ( x ) + 2 ε h ( x ) · [ h ( x ) + h ( z ) − h ( lca ( x, z )) − h ( lca ( y , z ))] > h ( y ) − h ( x ) + 2 ε h ( x ) · [ h ( x ) − h ( lca ( x, y ))] (48) = d ε ( x, y ) . Where in (48) w e us ed the fact that h ( z ) > h ( lca ( x, z )) + h ( lca ( y , z )) − h ( lca ( x, y )), whic h is true since h ( lca ( x, y )) > min { h ( lca ( x, z )) , h ( lca ( y , z )) } . It r emains to deal with the case h ( x ) 6 h ( z ) 6 h ( y ). In this case d ε ( x, z ) + d ε ( z , y ) = h ( y ) − h ( x ) + 2 ε h ( x ) · [ h ( x ) − h ( lca ( x, z ))] + 2 ε h ( z ) · [ h ( z ) − h ( lca ( y , z ))] > h ( y ) − h ( x ) + 2 ε h ( x ) · [ h ( x ) − h ( lca ( x, z ))] + 2 h ( x ) h ( z ) ε h ( x ) · [ h ( z ) − h ( lca ( y , z ))] = h ( y ) − h ( x ) + 2 ε h ( x ) ·  2 h ( x ) − h ( lca ( x, z )) − h ( x ) h ( z ) h ( lca ( y , z ))  > h ( y ) − h ( x ) + 2 ε h ( x ) · [ h ( x ) − h ( lca ( x, y ))] (49) = d ε ( x, y ) , where (49) is equiv alen t to the inequalit y h ( x ) > h ( lca ( x, z )) + h ( x ) h ( z ) h ( lca ( y , z )) − h ( lca ( x, y )) . (50) T o pro ve (50), note that it is tru e if h ( lca ( x, y )) > h ( lca ( x, z )), since clearly h ( lca ( y , z )) 6 h ( z ). If, on the other h and, h ( lca ( x, y )) < h ( lca ( x, z )) then u sing the assum ption that h ( z ) > h ( x ) it is enough to sh o w th at h ( x ) > h ( lca ( x, z )) + h ( lca ( y , z )) − h ( lca ( x, y )). Necessarily h ( lca ( x, y )) = h ( lca ( y , z )), so that the required inequalit y follo ws fr om the f act th at h ( x ) > h ( lca ( x, z )). 22 6.2 Geometry of H -trees 6.2.1 Classification of a pproximate midp oin ts F rom no w on we will alw ays assu me th at ε = { ε n } ∞ n =0 satisfies for all n ∈ N , ε n > ε n +1 > 0 and ( n + 1) ε n +1 > nε n . W e recall the imp ortan t concept of appr oximate midp oints wh ic h is u sed frequent ly in nonlinear functional analysis (see [4] and the references therein). Definition 6.2 (Appro ximate midp oin ts) . L e t ( X, d X ) b e a metric sp ac e a nd δ ∈ (0 , 1) . F or x, y ∈ X the set of δ -appr oxima te midp oints of x and z is define d as Mid( x, z , δ ) =  y ∈ X : max { d X ( x, y ) , d X ( y , z ) } 6 1 + δ 2 · d X ( x, z )  . F rom n o w on, w henev er w e refer to the set Mid( x, z , δ ), the under lyin g metric will alwa ys b e understo o d to b e d ε . In wh at follo ws, given η > 0 w e shall say that tw o sequen ces ( u 1 , . . . , u n ) and ( v 1 , . . . , v n ) of ve rtices in B ∞ are η -near if f or ev er y j ∈ { 1 , . . . , n } w e hav e d ε ( u j , v j ) 6 η . W e shall also requir e the follo wing terminology: Definition 6.3. An or der e d triple ( x, y , z ) of vertic es in B ∞ wil l b e c al le d a path-t yp e configuration if h ( z ) 6 h ( y ) 6 h ( x ) , x is a desc endant of y , and h ( lca ( z , y )) < h ( y ) . The triple ( x, y , z ) wil l b e c al le d a ten t-t yp e configuration if h ( y ) 6 h ( z ) , y is a desc endant of x , and h ( lca ( x, z )) < h ( x ) . These sp e cial c onfigur ations ar e describ e d in Figur e 3. x y z P ath-t yp e T en t-type x y z Figure 3: A schematic description of p ath-typ e and tent-typ e c onfigur ations. The follo wing useful theorem will b e u sed extensiv ely in the ensuing argu m en ts. Its pro of will b e b rok en d own in to s ev eral elemen tary lemmas. Theorem 6.4 . A ssume that δ ∈ (0 , 1 16 ) , and the se quenc e ε = { ε n } ∞ n =0 satisfies ε n < 1 4 for al l n ∈ N . L et x, y , z ∈ ( B ∞ , d ε ) b e such that y ∈ Mid( x, z , δ ) . Then ei ther ( x, y , z ) or ( z , y , x ) is 3 δ d ε ( x, z ) -ne ar a p ath-typ e or tent-typ e c onfigur ation. In w h at follo ws , giv en a vertex v ∈ B ∞ w e d enote the subtree ro oted at v by T v . Lemma 6.5. Assume that ε n 6 1 2 for al l n . Fix a ∈ B ∞ and let u, v ∈ B ∞ b e i ts childr en. F or every x, z ∈ T u such that h ( x ) > h ( z ) c onsider the function D x,z : { a } ∪ T v → [0 , ∞ ) define d by D x,z ( y ) = d ε ( x, y ) + d ε ( z , y ) . Fix an arbitr ary ve rtex w ∈ T v such that h ( w ) = h ( z ) . Then for every y ∈ T v we have D x,z ( y ) > D x,z ( w ) . 23 Pr o of. By the definition of d ε w e h a ve D x,z ( y ) = Q ( h ( y )) where Q ( k ) = max { h ( x ) , k } + max { k , h ( z ) } − min { h ( x ) , k } − min { k , h ( z ) } + 2 ε min { h ( x ) ,k } [min { h ( x ) , k } − h ( a )] + 2 ε min { k ,h ( z ) } [min { k , h ( z ) } − h ( a )] . The required result will follo w if we sh ow that Q is non-increasing on { h ( a ) , h ( a ) + 1 , . . . , h ( z ) } and non-decreasing on { h ( z ) , h ( z ) + 1 , . . . } . If k ∈ { h ( a ) , h ( a ) + 1 , . . . , h ( z ) − 1 } th en Q ( k + 1) − Q ( k ) = − 2 + 4 ε k +1 [ k + 1 − h ( a )] − 4 ε k [ k − h ( a )] 6 − 2 + 4 ε k [ k + 1 − h ( a )] − 4 ε k [ k − h ( a )] = − 2(1 − 2 ε k ) 6 0 . If k ∈ { h ( x ) , h ( x ) + 1 , . . . } th en Q ( k + 1) − Q ( k ) = 2, and if k ∈ { h ( z ) , . . . , h ( x ) − 1 } then Q ( k + 1) − Q ( k ) = 2 [ ( k + 1) ε k +1 − k ε k ] + 2 h ( a )[ ε k − ε k +1 ] > 0 . This completes the p r o of of Lemma 6.5. Lemma 6.6 . Assume that ε n < 1 2 for al l n ∈ N . Fix δ ∈  0 , 1 3  and x, y , z ∈ B ∞ such that h ( x ) > h ( z ) , y ∈ Mid( x, z , δ ) and h ( lca ( x, z )) > h ( lca ( x, y )) . Then h ( z ) + 1 − 3 δ 2 d ε ( x, z ) 6 h ( y ) < h ( x ) 6 h ( y ) + 1 + 3 δ 1 − 3 δ [ h ( y ) − h ( z )] . Mor e over, if y ′ ∈ B ∞ is the p oint on the se gment joining x and lca ( x, y ) such that h ( y ′ ) = h ( y ) then d ε ( y , y ′ ) 6 δ d ε ( x, z ) . Thus ( x, y ′ , z ) is a p ath-typ e c onfigur ation which is δ d ε ( x, z ) -ne ar ( x, y , z ) Pr o of. W rite a = lca ( x, y ). If u, v are the t w o c h ildren of a , then without loss of generalit y x , z ∈ T u and y ∈ T v . Let w ∈ T v b e s uc h that h ( w ) = h ( z ). y z a x w y ′ u v By Lemm a 6.5, d ε ( x, y ) + d ε ( z , y ) > d ε ( x, w ) + d ε ( z , w ) = h ( x ) − h ( z ) + 4 ε h ( z ) [ h ( z ) − h ( a )] > 2 d ε ( x, z ) − [ h ( x ) − h ( z )] . (51) On th e other hand, since y ∈ Mid( x, z , δ ), w e ha ve that d ε ( x, y ) + d ε ( z , y ) 6 (1 + δ ) d ε ( x, z ) . Additionally , b y the d efinition of d ε w e kn o w that if h ( y ) 6 h ( z ) then 1 + δ 2 d ε ( x, z ) > d ε ( x, y ) > h ( x ) − h ( y ) > h ( x ) − h ( z ) . Com bining th ese observ ations with (51) we get that (1 + δ ) d ε ( x, z ) > 2 d ε ( x, z ) − 1 + δ 2 d ε ( x, z ) , (52) whic h is a con tradiction since δ < 1 3 . Therefore h ( y ) > h ( z ). If h ( y ) > h ( x ) then 1 + δ 2 d ε ( x, z ) > d ε ( z , y ) > h ( y ) − h ( z ) > h ( x ) − h ( z ) , 24 so that w e arrive at a contradict ion as in (52). W e hav e th us s h o w n that h ( z ) < h ( y ) < h ( x ). No w, since y ∈ Mid( x, z , δ ), h ( x ) − h ( z ) + 2 ε h ( y ) [ h ( y ) − h ( a )] + 2 ε h ( z ) [ h ( z ) − h ( a )] = d ε ( x, y ) + d ε ( z , y ) 6 (1 + δ ) d ε ( x, z ) = (1 + δ )  h ( x ) − h ( z ) + 2 ε h ( z ) [ h ( z ) − h ( a )]  . Th us, letti ng y ′ b e th e p oin t on the segmen t joining x and a suc h that h ( y ′ ) = h ( y ), we see that d ε ( y , y ′ ) = 2 ε h ( y ) [ h ( y ) − h ( a )] 6 δ  h ( x ) − h ( z ) + 2 ε h ( z ) [ h ( z ) − h ( a )]  = δ d ε ( x, z ) , Moreo ve r 1 − δ 2 d ε ( x, z ) 6 d ε ( y , z ) = h ( y ) − h ( z ) + 2 ε h ( z ) h ( z ) − 2 ε h ( z ) h ( a ) 6 h ( y ) − h ( z ) + 2 ε h ( y ) h ( y ) − 2 ε h ( y ) h ( a ) 6 h ( y ) − h ( z ) + δ d ε ( x, z ) . Th us h ( y ) − h ( z ) > 1 − 3 δ 2 d ε ( x, z ) . (53) Hence, 2 1 − 3 δ [ h ( y ) − h ( z )] (53) > d ε ( x, z ) = [ h ( x ) − h ( y )] + [ h ( y ) − h ( z )] . It follo w s that h ( x ) − h ( y ) 6 1 + 3 δ 1 − 3 δ [ h ( y ) − h ( z )] . This completes the p r o of of Lemma 6.6. Lemma 6.7. A ssu me that ε n < 1 4 for al l n ∈ N . Fix δ ∈  0 , 1 16  and assume that x, y , z ∈ B ∞ ar e distinct vertic es such that l ca ( x, y ) = lca ( x, z ) , and y ∈ Mid( x, z , δ ) . Then either ( x, y , z ) or ( z , y , x ) is 3 δ d ε ( x, z ) -ne ar a p ath-typ e or tent-typ e c onfigur ation. Pr o of. Denote a = lca ( x, y ). Our assumption implies that h ( lca ( z , y )) > h ( a ). W e p erform a case analysis on the relativ e heigh ts of x, y , z . Assume fi rst that h ( x ) 6 h ( y ). y z y ′ a lca ( y , z ) x If h ( x ) 6 h ( y ) 6 h ( z ) then (1 + δ ) d ε ( x, z ) > d ε ( x, y ) + d ε ( y , z ) = h ( y ) − h ( x ) + 2 ε h ( x ) [ h ( x ) − h ( a )] + h ( z ) − h ( y ) + 2 ε h ( y ) [ h ( y ) − h ( lca ( z , y ))] = d ε ( x, z ) + 2 ε h ( y ) [ h ( y ) − h ( lca ( z , y ))] . (54) Let y ′ b e the p oint on the path from lca ( y , z ) to z such that h ( y ′ ) = h ( y ). Then (54) implies that d ε ( y , y ′ ) = 2 ε h ( y ) [ h ( y ) − h ( lca ( z , y ))] 6 δ d ε ( x, z ) . Th us the triple ( z , y ′ , x ) is a configuration of path-t yp e w hic h is δ d ε ( x, z )-near ( z , y , x ). 25 a lca ( y , z ) x y z If h ( x ) 6 h ( z ) 6 h ( y ) then since 1 + δ 2 d ε ( x, z ) > d ε ( x, y ) = h ( y ) − h ( x ) + 2 ε h ( x ) [ h ( x ) − h ( a )] and d ε ( x, z ) = h ( z ) − h ( x ) + 2 ε h ( x ) [ h ( x ) − h ( a )] we dedu ce that − 1 − δ 2 d ε ( x, z ) > d ε ( x, y ) − d ε ( x, z ) = h ( y ) − h ( z ) > 0 . It follo w s that x = z , in con tr ad iction to our assumption. a lca ( y , z ) x y z z ′ y ′ If h ( z ) < h ( x ) th en let z ′ b e the p oint on the segmen t joinin g a and y su c h that h ( z ′ ) = h ( z ). W e th us ha ve that d ε ( z , z ′ ) = 2 ε h ( z ) [ h ( z ) − h ( lca ( y , z ))] = d ε ( z , y ) − [ h ( y ) − h ( z )] . Moreo ve r, 2 ε h ( z ) [ h ( lca ( z , y )) − h ( a )] = d ε ( x, z ) − [ h ( x ) − h ( z )] − 2 ε h ( z ) [ h ( z ) − h ( lca ( z , y ))] > 2 1 + δ d ε ( z , y ) − [ h ( x ) − h ( z )] − 2 ε h ( z ) [ h ( z ) − h ( lca ( z , y ))] = 2 1 + δ  h ( y ) − h ( z ) + 2 ε h ( z ) [ h ( z ) − h ( lca ( y , z ))]  −  h ( y ) − h ( z ) + 2 ε h ( z ) [ h ( z ) − h ( lca ( y , z ))]  + [ h ( y ) − h ( x )] = 1 − δ 1 + δ d ε ( y , z ) + [ h ( y ) − h ( x )] > 1 − δ 1 + δ · 1 − δ 2 d ε ( x, z ) + [ h ( y ) − h ( x )] >  1 − δ 1 + δ  2 d ε ( x, y ) + [ h ( y ) − h ( x )] = d ε ( x, y ) + [ h ( y ) − h ( x )] − 4 δ (1 + δ ) 2 d ε ( x, y ) = 2[ h ( y ) − h ( x )] + 2 ε h ( x ) [ h ( x ) − h ( a )] − 4 δ (1 + δ ) 2 d ε ( x, y ) > 2[ h ( y ) − h ( x )] + 2 ε h ( z ) h ( z ) − 2 ε h ( z ) h ( a ) − 2 δ 1 + δ d ε ( x, z ) . Th us 2 δ 1 + δ d ε ( x, z ) > 2[ h ( y ) − h ( x )] + 2 ε h ( z ) [ h ( z ) − h ( lca ( y , z ))] = 2[ h ( y ) − h ( x )] + d ε ( z , z ′ ) . (55) Let y ′ b e the p oint on the path from a to y such that h ( y ′ ) = h ( x ). It follo w s from (55) that the triple ( z ′ , y ′ , x ) is a configur ation of ten t-type wh ic h is 2 δ d ε ( x, z )-near ( z , y , x ). This completes the pr o of of L emma 6.7 when h ( x ) 6 h ( y ). The case h ( x ) > h ( y ) is prov ed analogously . Here are the details. 26 a lca ( y , z ) x y z y ′ x ′ Assume fi rst of all that h ( z ) > h ( x ) > h ( y ). Th en , d ε ( x, z ) > 2 1 + δ d ε ( z , y ) = 2 d ε ( z , y ) − 2 δ 1 + δ d ε ( z , y ) = 2  h ( z ) − h ( y ) + 2 ε h ( y ) [ h ( y ) − h ( lca ( z , y ))]  − 2 δ 1 + δ d ε ( z , y ) > 2  h ( z ) − h ( y ) + 2 ε h ( y ) [ h ( y ) − h ( lca ( z , y ))]  − δ d ε ( x, z ) . (56) On th e other hand, since h ( x ) > h ( y ), d ε ( x, z ) = h ( z ) − h ( x ) + 2 ε h ( x ) [ h ( x ) − h ( a )] 6 h ( z ) − h ( x ) + 2 ε h ( y ) [ h ( x ) − h ( a )] =  h ( x ) − h ( y ) + 2 ε h ( y ) [ h ( y ) − h ( a )]  + h ( y ) + h ( z ) − 2 h ( x ) + 2 ε h ( y ) [ h ( x ) − h ( y )] = d ε ( x, y ) + h ( y ) + h ( z ) − 2 h ( x ) + 2 ε h ( y ) [ h ( x ) − h ( y )] 6 1 + δ 1 − δ d ε ( y , z ) + h ( y ) + h ( z ) − 2 h ( x ) + 2 ε h ( y ) [ h ( x ) − h ( y )] =  h ( z ) − h ( y ) + 2 ε h ( y ) [ h ( y ) − h ( lca ( z , y ))]  + 2 δ 1 − δ d ε ( y , z ) + h ( y ) + h ( z ) − 2 h ( x ) + 2 ε h ( y ) [ h ( x ) − h ( y )] 6 2[ h ( z ) − h ( x )] + 2 ε h ( y ) [ h ( x ) − h ( lca ( z , y ))] + 1+ δ 1 − δ δ d ε ( x, z ) . Com bining th is b ound with (56), an d canceling term s , giv es 2 δ 1 − δ d ε ( x, z ) > 2[ h ( x ) − h ( y )] − 4 ε h ( y ) [ h ( x ) − h ( y )] + 2 ε h ( y ) [ h ( x ) − h ( lca ( z , y ))] > 2  1 − 2 ε h ( y )  [ h ( x ) − h ( y )] + 2 ε h ( y ) [ h ( x ) − h ( lca ( z , y ))] > [ h ( x ) − h ( y )] + 2 ε h ( y ) [ h ( x ) − h ( lca ( z , y ))] , (57) where we us ed the fact that ε h ( y ) < 1 4 . Let x ′ b e the p oin t on the path f rom x to a such that h ( x ′ ) = h ( y ), and let y ′ b e the p oint on the path fr om a to z suc h th at h ( y ′ ) = h ( y ). T hen by (57) d ε ( x, x ′ ) = h ( x ) − h ( y ) 6 3 δ d ε ( x, z ) and d ε ( y , y ′ ) = 2 ε h ( y ) [ h ( y ) − h ( lca ( z , y ))] 6 2 ε h ( y ) [ h ( x ) − h ( lca ( z , y ))] 6 3 δ d ε ( x, z ) . Th us the triple ( z , y ′ , x ′ ) is a configuration of path-t yp e whic h is 3 δ d ε ( x, z )-near ( z , y , x ). x a lca ( y , z ) y z If h ( x ) > h ( z ) > h ( y ) then h ( z ) − h ( y ) + 2 ε h ( y ) [ h ( y ) − h ( lca ( z , y ))] = d ε ( z , y ) > 1 − δ 1 + δ d ε ( x, y ) = d ε ( x, y ) − 2 δ 1 + δ d ε ( x, y ) = h ( x ) − h ( y )+2 ε h ( y ) [ h ( y ) − h ( a )] − 2 δ 1 + δ d ε ( x, y ) . Canceling terms w e see that 2 δ 1 + δ d ε ( x, y ) > h ( x ) − h ( z ) + 2 ε h ( y ) [ h ( lca ( z , y )) − h ( a )] = h ( x ) − h ( z ) + 2 ε h ( y ) [ h ( z ) − h ( a )] − 2 ε h ( y ) [ h ( z ) − h ( lca ( z , y ))] > h ( x ) − h ( z ) + 2 ε h ( z ) [ h ( z ) − h ( a )] − 2 ε h ( y ) [ h ( z ) − h ( lca ( z , y ))] = d ε ( x, z ) − 2 ε h ( y ) [ h ( z ) − h ( lca ( z , y ))] 27 > 2 d ε ( z , y ) − 2 δ 1 + δ d ε ( x, z ) − 2 ε h ( y ) [ h ( z ) − h ( lca ( z , y ))] = 2  h ( z ) − h ( y ) + 2 ε h ( y ) [ h ( y ) − h ( lca ( z , y ))]  − 2 δ 1 + δ d ε ( x, z ) − 2 ε h ( y ) [ h ( z ) − h ( lca ( z , y ))] = 2  1 − 2 ε h ( y )  [ h ( z ) − h ( y )] + 2 ε h ( y ) [ h ( z ) − h ( lca ( z , y ))] − 2 δ 1 + δ d ε ( x, z ) > [ h ( z ) − h ( y )] + 2 ε h ( y ) [ h ( z ) − h ( lca ( z , y ))] − 2 δ 1 + δ d ε ( x, z ) = d ε ( z , y ) − 2 δ 1 + δ d ε ( x, z ) >  1 − δ 2 − 2 δ 1 + δ  d ε ( x, z ) , whic h is a con tradiction since δ < 1 16 . y z a lca ( y , z ) x z ′ The only remaining case is w h en h ( x ) > h ( y ) > h ( z ). I n th is case we pro ceed as follo ws. d ε ( x, z ) = h ( x ) − h ( z ) + 2 ε h ( z ) [ h ( z ) − h ( a )] = d ε ( y , z ) + [ h ( x ) − h ( y )] + 2 ε h ( z ) [ h ( lca ( y , z )) − h ( a )] > d ε ( x, y ) − 2 δ 1 + δ d ε ( x, y ) + [ h ( x ) − h ( y )] + 2 ε h ( z ) [ h ( lca ( y , z )) − h ( a )] > h ( x ) − h ( y ) + 2 ε h ( y ) [ h ( y ) − h ( a )] − δ d ε ( x, z ) + [ h ( x ) − h ( y )] + 2 ε h ( z ) [ h ( lca ( y , z )) − h ( a )] > 2[ h ( x ) − h ( y )] + 2 ε h ( z ) [ h ( z ) − h ( a )] + 2 ε h ( z ) [ h ( lca ( y , z )) − h ( a )] − δ d ε ( x, z ) = 2[ h ( x ) − h ( y )] + 4 ε h ( z ) [ h ( z ) − h ( a )] − 2 ε h ( z ) [ h ( z ) − h ( lca ( y , z ))] − δd ε ( x, z ) = 2 d ε ( x, z ) − 2 d ε ( z , y ) + 2 ε h ( z ) [ h ( z ) − h ( lca ( y , z ))] − δ d ε ( x, z ) > (1 − 2 δ ) d ε ( x, z ) + 2 ε h ( z ) [ h ( z ) − h ( lca ( y , z ))] . (58) Let z ′ b e the p oint on the path from a to y su c h that h ( z ′ ) = h ( z ). Then d ε ( z , z ′ ) = 2 ε h ( z ) [ h ( z ) − h ( lca ( y , z ))] (58) 6 2 δ d ε ( x, z ) . Therefore the triple ( z ′ , y , x ) is of tent-t yp e and is 2 δ d ε ( x, z )-near ( z , y , x ). The pro of of Lemma 6.7 is complete. Pr o of of The or em 6.4. It remains to c h ec k that for ev ery x, y , z ∈ B ∞ suc h that y ∈ Mid( x, z , δ ), at least one of the triples ( x, y , z ) or ( z , y , x ) s atisfies the conditions of Lemm a 6.6 or Lemma 6.7 . Indeed, if h ( lca ( x, y )) = h ( lca ( x, z )) then lca ( x, y ) = lca ( x, z ), so Lemma 6.7 applies. If h ( lca ( x, y )) < h ( lca ( x, z )) th en l ca ( z , y ) = lca ( x, z ), so Lemma 6.7 applies to the triple ( z , y , x ). If h ( lca ( x, y )) < h ( lca ( x, z )) then lca ( x, y ) = l ca ( z , y ), and so h ( lca ( x, z )) > h ( lca ( z , y )). Hence Lemma 6.6 applies to either the triple ( x, y , z ) or the trip le ( z , y , x ). 28 W e end this sub section with a short discussion on the distance b et ween ten t-type and path-t y p e configurations. It tu rns out that when ε h ≪ δ , a δ -midp oin t configuration ( x, y , z ) can b e close to a p ath-t yp e configur ation, and at th e same time the rev ersed triple ( z , y , x ) close to a ten t-t y p e configuration (or vice versa). Ho w ev er , it is easy to see that this is the only “closeness” p ossible. Lemma 6.8. Fix x, y , z ∈ B ∞ with x 6 = y . Then the fol lowing statements ar e imp ossible: 1. ( x, y , z ) is 1 5 d ε ( x, y ) -ne ar a p ath-typ e c onfigur ation and a tent-typ e c onfigur ation. 2. ( x, y , z ) is 1 11 d ε ( x, y ) -ne ar a p ath-typ e c onfigur ation and ( z , y , x ) is 1 11 d ε ( x, y ) -ne ar a p ath-typ e c onfigur ation. 3. ( x, y , z ) is 1 11 d ε ( x, y ) -ne ar a tent-typ e c onfigur ation and ( z , y , x ) is 1 11 d ε ( x, y ) -ne ar a tent-typ e c onfigur ation. Pr o of. F or case 1 of Lemma 6.8, assum e for contradicti on that ( x, y , z ) is 1 5 d ε ( x, y )-near a path- t yp e configu r ation ( a 1 , b 1 , c 1 ), and also 1 5 d ε ( x, y )-near a ten t-type configur ation ( α 1 , β 1 , γ 1 ). By the definitions of p ath-t yp e and ten t-t yp e configur ations, a 1 is a descendant of b 1 and β 1 is a descend an t of α 1 . Hence, h ( a 1 ) − h ( b 1 ) = d ε ( a 1 , b 1 ) > d ε ( x, y ) − d ε ( x, a 1 ) − d ε ( y , b 1 ) > 3 5 d ε ( x, y ) , (59) and h ( β 1 ) − h ( α 1 ) = d ε ( α 1 , β 1 ) > d ε ( x, y ) − d ε ( x, α 1 ) − d ε ( y , β 1 ) > 3 5 d ε ( x, y ) . (60) By su mming (59) and (60) w e s ee that, 4 5 d ε ( x, y ) > d ε ( a 1 , x ) + d ε ( x, α 1 ) + d ε ( b 1 , y ) + d ε ( y , β 1 ) > d ε ( a 1 , α 1 ) + d ε ( b 1 , β 1 ) > h ( a 1 ) − h ( α 1 ) + h ( β 1 ) − h ( b 1 ) (59) ∧ (60) > 6 5 d ε ( x, y ) , a contradict ion. F or case 2 of Lemma 6.8, assume for con tradiction that ( x, y , z ) is 1 11 d ε ( x, y )-near a path-type configuration ( a 2 , b 2 , c 2 ), and also ( z , y , x ) is 1 11 d ε ( x, y )-near a path-t yp e confi guration ( α 2 , β 2 , γ 2 ). By the definitions of path-t yp e and ten t-type confi gurations, a 2 is a descendant of b 2 and h ( β 2 ) > h ( γ 2 ). Hence, h ( a 2 ) − h ( b 2 ) = d ε ( a 2 , b 2 ) > d ε ( x, y ) − d ε ( x, a 2 ) − d ε ( y , b 2 ) > 9 11 d ε ( x, y ) , (61) and h ( β 2 ) − h ( γ 2 ) + 2 ε h ( γ 2 ) [ h ( γ 2 ) − h ( lca ( β 2 , γ 2 ))] = d ε ( β 2 , γ 2 ) > d ε ( x, y ) − d ε ( x, γ 2 ) − d ε ( y , β 2 ) > 9 11 d ε ( x, y ) . (62) 29 By su mming (61) and (62) w e s ee that 17 11 d ε ( x, y ) > d ε ( a 2 , x ) + d ε ( x, γ 2 ) + d ε ( b 2 , y ) + d ε ( y , β 2 ) + d ε ( β 2 , y ) + d ε ( x, y ) + d ε ( x, γ 2 ) > d ε ( a 2 , γ 2 ) + d ε ( b 2 , β 2 ) + d ε ( β 2 , γ 2 ) >  h ( a 2 ) − h ( γ 2 )  +  h ( β 2 ) − h ( b 2 )  + 2 ε h ( γ 2 ) [ h ( γ 2 ) − h ( lca ( β 2 , γ 2 ))] (61) ∧ (62) > 18 11 d ε ( x, y ) , a contradict ion. F or case 3 of L emm a 6.8, assu m e for con tradiction that ( x, y , z ) is 1 11 d ε ( x, y )-near a tent-t yp e configuration ( a 3 , b 3 , c 3 ), and also ( z , y , x ) is 1 11 d ε ( x, y )-near a tent -t yp e configur ation ( α 3 , β 3 , γ 3 ). Then b 3 is a descendan t of a 3 and h ( γ 3 ) > h ( β 3 ). Hence, h ( b 3 ) − h ( a 3 ) = d ε ( a 3 , b 3 ) > d ε ( x, y ) − d ε ( x, a 3 ) − d ε ( y , b 3 ) > 9 11 d ε ( x, y ) , (63) and h ( γ 3 ) − h ( β 3 ) + 2 ε h ( β 3 ) [ h ( β 3 ) − h ( lca ( β 3 , γ 3 ))] = d ε ( β 3 , γ 3 ) > d ε ( x, y ) − d ε ( x, γ 3 ) − d ε ( y , β 3 ) > 9 11 d ε ( x, y ) . (64) Hence, 17 11 d ε ( x, y ) > d ε ( b 3 , y ) + d ε ( y , β 3 ) + d ε ( x, α 3 ) + d ε ( x, γ 3 ) + d ε ( β 3 , y ) + d ε ( x, y ) + d ε ( x, γ 3 ) > d ε ( b 3 , β 3 ) + d ε ( a 3 , γ 3 ) + d ε ( β 3 , γ 3 ) >  h ( b 3 ) − h ( β 3 )  +  h ( γ 3 ) − h ( a 3 )  + 2 ε h ( β 3 ) [ h ( β 3 ) − h ( lca ( β 3 , γ 3 ))] (63) ∧ (64) > 18 11 d ε ( x, y ) , a contradict ion. 6.2.2 Classification of a pproximate f orks W e b egin with three “stitc hing lemmas” that roughly say that giv en three p oint s x, x ′ , y ∈ ( B ∞ , d ε ) suc h that x ′ is near x , there exists y ′ near y s u c h that d ε ( x ′ , y ′ ) is close to d ε ( x, y ), and y ′ relates to x ′ in B ∞ “in the same wa y” that y relates x in B ∞ . Lemma 6.9. L et x, x ′ , y , y ′ ∈ B ∞ b e su c h that y is an anc estor of x , and y ′ is an anc estor of x ′ satisfying h ( x ) − h ( y ) = h ( x ′ ) − h ( y ′ ) . Then d ε ( y , y ′ ) 6 d ε ( x, x ′ ) . Pr o of. Assume without loss of generalit y that h ( x ) > h ( x ′ ). So, d ε ( x, x ′ ) = h ( x ) − h ( x ′ ) + 2 ε h ( x ′ ) [ h ( x ′ ) − h ( lca ( x, x ′ ))] . Note that h ( lca ( y , y ′ )) = min { h ( y ′ ) , h ( lca ( x, x ′ )) } . Hence, d ε ( y , y ′ ) = h ( y ) − h ( y ′ ) + 2 ε h ( y ′ ) [ h ( y ′ ) − h ( lca ( y , y ′ ))] = h ( x ) − h ( x ′ ) + 2 ε h ( y ′ ) [ h ( y ′ ) − min { h ( y ′ ) , h ( lca ( x, x ′ )) } ] = h ( x ) − h ( x ′ ) + 2 ε h ( y ′ ) max { 0 , h ( y ′ ) − h ( lca ( x, x ′ )) } . (65) 30 If the maxim um in (65) is 0, then d ε ( y , y ′ ) = h ( x ) − h ( x ′ ) 6 d ε ( x, x ′ ) . If the maxim um in (65) equals h ( y ′ ) − h ( lca ( x, x ′ )), then d ε ( y , y ′ ) = h ( x ) − h ( x ′ ) + 2 ε h ( y ′ ) [ h ( y ′ ) − h ( lca ( x, x ′ ))] 6 h ( x ) − h ( x ′ ) + 2 ε h ( x ′ ) [ h ( x ′ ) − h ( lca ( x, x ′ ))] = d ε ( x, x ′ ) , (66 ) where in (66) we used the fact that the sequence { ε n ( n − a ) } ∞ n =0 is nond ecreasing for all a > 0. Lemma 6.10. L e t x, x ′ , y ∈ B ∞ b e such that h ( y ) 6 h ( x ) . Then ther e e xists y ′ ∈ B ∞ which satisfies h ( y ′ ) − h ( x ′ ) = h ( y ) − h ( x ) , d ε ( y , y ′ ) 6 d ε ( x, x ′ ) , (67) and d ε ( x, y ) − 2 d ε ( x, x ′ ) 6 d ε ( y ′ , x ′ ) 6 d ε ( x, y ) + 2 d ε ( x, x ′ ) . (68) Pr o of. Note that (68) follo ws from (67) b y the triangle in equalit y . Assu me fir st that h ( x ) > h ( x ′ ). In this case c ho ose y ′ to b e an ancesto r of y satisfying h ( y ) − h ( y ′ ) = h ( x ) − h ( x ′ ). Th en, d ε ( y , y ′ ) = h ( y ) − h ( y ′ ) = h ( x ) − h ( x ′ ) 6 d ε ( x, x ′ ) . W e n ext assume that h ( x ) < h ( x ′ ). If h ( lca ( x, x ′ )) 6 = h ( lca ( x, y )) then c h o ose y ′ to b e an arbitrary descendan t of y such that h ( y ′ ) − h ( y ) = h ( x ′ ) − h ( x ). As b efore, we conclude that d ε ( y , y ′ ) = h ( y ′ ) − h ( y ) = h ( x ′ ) − h ( x ) 6 d ε ( x, x ′ ). It remains to deal with th e case h ( x ′ ) > h ( x ) and h ( lca ( x, y )) = h ( lca ( x ′ , x )), which also implies that h ( lca ( x ′ , y )) > h ( lca ( x, y )). In this case, we choose y ′ to b e an arb itrary p oin t on a branch con taining b oth lca ( x, y ) and x , s u c h that h ( y ′ ) − h ( y ) = h ( x ′ ) − h ( x ). Th en lca ( y ′ , y ) = lca ( x, x ′ ), and therefore, d ε ( y , y ′ ) = h ( y ′ ) − h ( y ) + 2 ε h ( y ) [ h ( y ) − h ( lca ( y , y ′ )] = h ( x ′ ) − h ( x ) + 2 ε h ( y ) [ h ( y ) − h ( lca ( x, x ′ ))] 6 h ( x ′ ) − h ( x ) + 2 ε h ( x ) [ h ( x ) − h ( lca ( x, x ′ ))] = d ε ( x, x ′ ) , pro ving (67) in the last remaining case. Lemma 6.11. L et x, x ′ , y ∈ B ∞ b e such that y is a desc endant of x . Then for any y ′ ∈ B ∞ which is a desc endant of x ′ and satisfying h ( y ′ ) − h ( x ′ ) = h ( y ) − h ( x ) , we have d ε ( y , y ′ ) 6 d ε ( x, x ′ ) + 2 ε min { h ( y ′ ) ,h ( y ) } [ h ( y ) − h ( x )] 6 d ε ( x, x ′ ) + 2 ε h ( y ) [ h ( y ) − h ( x ) + d ε ( x, x ′ )] . Pr o of. Note that h ( lca ( y , y ′ )) > h ( lca ( x, x ′ )). Ass ume first that h ( x ′ ) > h ( x ). Then, d ε ( y , y ′ ) = h ( y ′ ) − h ( y ) + 2 ε h ( y ) [ h ( y ) − h ( lca ( y , y ′ ))] = h ( x ′ ) − h ( x ) + 2 ε h ( y ) [ h ( x ) − h ( lca ( y , y ′ ))] + 2 ε h ( y ) [ h ( y ) − h ( x )] 6 h ( x ′ ) − h ( x ) + 2 ε h ( y ) [ h ( x ) − h ( lca ( x, x ′ )] + 2 ε h ( y ) [ h ( y ) − h ( x )] 6 d ε ( x, x ′ ) + 2 ε h ( y ) [ h ( y ) − h ( x )] . 31 When h ( x ′ ) < h ( x ), we similarly obtain the b ound: d ε ( y , y ′ ) = h ( y ) − h ( y ′ ) + 2 ε h ( y ′ ) [ h ( y ′ ) − h ( lca ( y , y ′ ))] = h ( x ) − h ( x ′ ) + 2 ε h ( y ′ ) [ h ( x ′ ) − h ( lca ( y , y ′ ))] + 2 ε h ( y ′ ) [ h ( y ′ ) − h ( x ′ )] 6 d ε ( x, x ′ ) + 2 ε h ( y ′ ) [ h ( y ) − h ( x )] . The last inequalit y in the statemen t of Lemma 6.11 is pro v ed by observin g that when h ( y ′ ) < h ( y ), ε h ( y ′ ) [ h ( y ) − h ( x )] = ε h ( y ′ ) [ h ( y ′ ) − h ( x ′ )] 6 ε h ( y ) [ h ( y ) − h ( x ′ )] 6 ε h ( y ) [ h ( y ) − h ( x ) + d ε ( x, x ′ )] . Definition 6.12. F or δ ∈ (0 , 1) and x, y , z , w ∈ B ∞ , the quadruple ( x, y , z , w ) is c al le d a δ -fork, if y ∈ Mid( x, z , δ ) ∩ Mid ( x, w, δ ) . δ -forks in H-trees can b e appro ximately classified using the approximat e classification of mid- p oint configu r ations of Section 6.2.1. W e ha v e four t yp es of mid p oint configurations (recall Fig- ure 3): • path-t yp e; denoted (P) in what follo ws, • rev ers e p ath-t yp e; denoted (p)—( x, y , z ) is of t yp e (p) iff ( z , y , x ) is of typ e (P), • ten t-t yp e; denoted (T), • rev ers e tent-t yp e; den oted (t)—( x, y , z ) is of type (t) iff ( z , y , x ) is of type (T). Th us, there are  5 2  = 10 p ossible δ -fork configurations in ( B ∞ , d ε ) (c h o ose tw o out of the fi v e sym b ols “P”,“p”,“T”,“t”,“ X”, w here “X” means “the same”). As we shall see, four of these p ossible configurations are imp ossible, t wo of them hav e large con traction of the p rongs of the forks, i.e., d ε ( z , w ) ≪ d ε ( x, y ), which imm ed iately implies large distortion, and the r est of the configurations are p roblematic in the sense that they are not muc h distorted fr om th e star K 1 , 3 (the metric d on four p oin ts p, q , r , s giv en by d ( p, q ) = d ( q , r ) = d ( q , s ) = 1 and d ( p, s ) = d ( p, r ) = d ( r, s ) = 2). T he 10 p ossible δ -fork configurations are summ arized in T able 1. Midp oint configuration Type (T k T) T yp e I (P k P) T yp e I I (p k T) T yp e I I I (p k t) T yp e I V (p k p) prongs con tracted (t k t) prongs con tracted (P k p) imp ossible (P k t) p ossible only as approximat e t yp e I I (P k T) imp ossible (t k T) imp ossible T able 1: The ten p ossible fork c onfigur ations. F or f uture r eference, we giv e names to th e four pr oblematic configurations: 32 Definition 6.13. F or η , δ ∈ (0 , 1) , a δ -fork ( x, y , z , w ) of ( B ∞ , d ε ) is c al le d • η -ne ar T yp e I (c onfigur ation (T k T)) in T able 1), if b oth ( x, y , z ) and ( x, y , w ) ar e η -ne ar tent-typ e c onfigur ations; • η -ne ar T yp e I I (c onfigur ation (P k P) in T able 1), if b oth ( x, y , z ) an d ( x, y , w ) ar e η - ne ar p ath-typ e c onfigur ations; • η -ne ar T yp e I I I (c onfigur ation (p k T) in T able 1), i f ( z , y , x ) is η -ne ar a p ath -typ e c onfigur a- tion and ( x, y , w ) is η -ne ar a tent-typ e c onfigur ation, or vic e versa; • η -ne ar T yp e I V (c onfigur ation (p k t) in T able 1), if ( z , y , x ) is η - ne ar a p ath-typ e c onfigur ation and ( w , y , x ) is η -ne ar a tent-typ e c onfigur ation, or vic e versa. A schematic description of the fou r problematic confi gu r ations is con tained in Figure 4. x y z w x y z w T yp e I T yp e I I T yp e I I I T yp e I V z y w x z y w x Figure 4: The four “pr oblematic” typ es of δ -forks. The follo wing lemma is the main result of this section. Lemma 6.14. Fix δ ∈  0 , 1 70  and a ssume that ε n < 1 4 for al l n ∈ N . If ( x, y , z , w ) is a δ - fork of ( B ∞ , d ε ) then ei ther it is 35 δ d ε ( x, y ) -ne ar o ne o f the typ es I , I I , I I I , I V , or we h ave d ε ( z , w ) 6 2(35 δ + ε h 0 ) d ε ( x, y ) , wher e h 0 = min { h ( x ) , h ( y ) , h ( z ) , h ( w ) } . Remark 6.15. One can strengthen th e s tatement of Lemm a 6.14 so that in the first case the fork ( x, y , z , w ) is O ( δ d ε ( x, y )) near another fork ( x ′ , y ′ , z ′ , w ′ ) whic h is of (i.e. 0-nea r) one of th e t yp es I , I I , I I I , I V . This statemen t is more complica ted to prov e, and since we do not actually need it in what follo w s, we opted to use a w eak er prop ert y whic h su ffices for our pu r p oses, yet simplifies (the already quite inv olv ed) pro of. The pro of of Lemma 6.14 p ro ceeds b y chec king that the cases mark ed in T able 1 as “imp ossible” or “prongs con tracted” are in deed so—see Figure 5 for a sc hematic description of the latter case. W e b egin with the ( p k p ) confi guration. Lemma 6.16. L et ( x, y , z , w ) b e a δ - fork of ( B ∞ , d ε ) and assume that b oth ( z , y , x ) , and ( w , y , x ) ar e η d ε ( x, y ) -ne ar p ath-typ e c onfigur ations. Then, assuming that max { δ, η } < 1 / 8 and ε n < 1 4 for al l n , we have d ε ( z , w ) 6  9 η + 6 δ + 2 ε h ( y )  d ε ( x, y ) . 33 x y z w z y w x Figure 5: The two c onfigur ations of δ - forks with lar ge c ontr action of the pr ongs. Pr o of. Let ( z ′ , y ′ , x ′ ) b e a p ath-t yp e configuration that is η d ε ( x, y )-near ( z , y , x ), and let ( w ′′ , y ′′ , x ′′ ) b e a path-t yp e configuration th at is η d ε ( x, y )-near ( w , y , x ). Without loss of generalit y assume that h ( y ′′ ) > h ( y ′ ). Let w ′ b e the descendant of y ′ satisfying h ( w ′ ) − h ( y ′ ) = h ( w ′′ ) − h ( y ′′ ) suc h th at w ′ is either an ancestor or an arbitrary descendant of z ′ . Note that h ( w ′ ) > h ( y ). Indeed, h ( w ′ ) = h ( y ′ ) + h ( w ′′ ) − h ( y ′′ ) = h ( y ′ ) + d ε ( w ′′ , y ′′ ) > h ( y ) − | h ( y ) − h ( y ′ ) | + d ε ( w, y ) − 2 η d ε ( x, y ) > h ( y ) − d ε ( y , y ′ ) + 1 − δ 1 + δ d ε ( x, y ) − 2 η d ε ( x, y ) > h ( y ) +  1 − δ 1 + δ − 3 η  d ε ( x, y ) > h ( y ) . By Lemm a 6.11 , d ε ( w ′ , w ′′ ) 6 d ε ( y ′ , y ′′ ) + 2 ε h ( w ′ )  h  w ′′  − h  y ′′  6 d ε ( y ′ , y ′′ ) + 2 ε h ( w ′ ) d ε ( y ′′ , w ′′ ) 6 2 η d ε ( x, y ) + 2 ε h ( y )  2 η + 1 + δ 1 − δ  d ε ( x, y ) . (69) Observe that  1 − δ 1 + δ − 2 η  d ε ( x, y ) 6 d ε ( z , y ) − 2 η d ε ( x, y ) 6 d ε ( z ′ , y ′ ) 6 d ε ( z , y ) + 2 η d ε ( x, y ) 6  1 + δ 1 − δ + 2 η  d ε ( x, y ) , (70) Since d ε ( w ′ , y ′ ) = d ε ( w ′′ , y ′′ ), we obtain similarly th e b ound s:  1 − δ 1 + δ − 2 η  d ε ( x, y ) 6 d ε ( w ′ , y ′ ) 6 d ε ( z , y ) + 2 η d ε ( x, y ) 6  1 + δ 1 − δ + 2 η  d ε ( x, y ) . (71) Hence d ε ( z ′ , w ′ ) =   d ε ( y ′ , z ′ ) − d ε ( y ′ , w ′ )   (70) ∧ (71) 6  4 δ 1 − δ 2 + 4 η  d ε ( x, y ) . (72) So, in conclusion, d ε ( z , w ) 6 d ε ( z , z ′ ) + d ε ( w, w ′′ ) + d ε ( w ′′ , w ′ ) + d ε ( z ′ , w ′ ) (69) ∧ (72) 6  8 η + 4 δ 1 − δ 2 + 2  2 η + 1 + δ 1 − δ  ε h ( y )  d ε ( x, y ) 6  9 η + 6 δ + 2 ε h ( y )  d ε ( x, y ) . 34 W e next consider the (t k t) confi gu r ation. Lemma 6.17. L et ( x, y , z , w ) b e a δ -fork of ( B ∞ , d ε ) . Assume that b oth ( z , y , x ) and ( w , y , x ) ar e η d ε ( x, y ) -ne ar tent- typ e c onfigur ations. Th en, assuming that max { δ , η } < 1 / 4 , we have d ε ( z , w ) 6 (8 η + 5 δ ) d ε ( x, y ) . Pr o of. Let ( z ′ , y ′ , x ′ ) b e a ten t-t yp e configuration that is η d ε ( x, y )-near ( z , y , x ), and let ( w ′′ , y ′′ , x ′′ ) b e a ten t-t yp e configuration th at is η d ε ( x, y )-near ( w, y , x ). Ass u me without loss of generalit y that h ( y ′′ ) − h ( w ′′ ) > h ( y ′ ) − h ( z ′ ). Let ˜ w b e a point on the path b et w een w ′′ and y ′′ suc h that h ( y ′′ ) − h ( ˜ w ) = h ( y ′ ) − h ( z ′ ). Th en, d ε ( w ′′ , ˜ w ) = h ( y ′′ ) − h ( w ′′ ) − ( h ( y ′ ) − h ( z ′ )) = d ε ( y ′′ , w ′′ ) − d ε ( y ′ , z ′ ) 6 d ε ( y , w ) − d ε ( y , z ) + 4 η d ε ( x, y ) 6  1 + δ 1 − δ − 1 − δ 1 + δ + 4 η  d ε ( x, y ) . (73) By Lemm a 6.9 we hav e d ε ( ˜ w , z ′ ) 6 d ε ( y ′ , y ′′ ) 6 2 η d ε ( x, y ). Hence we conclude that d ε ( y , z ) 6 d ε ( z , z ′ ) + d ε ( ˜ w , z ′ ) + d ε ( ˜ w , w ′′ ) + d ε ( w ′′ , w ) (73) 6  4 δ 1 − δ 2 + 8 η  d ε ( x, y ) . Lemma 6.18. L et ( x, y , z , w ) b e a δ -fork of B ∞ . Assume that ( x, y , z ) is η d ε ( x, y ) -ne ar a p ath-typ e c onfigur ation. Assume also that δ < 1 / 30 , η < 1 / 10 , and ε n < 1 / 4 for al l n . Then ( x, y , w ) is (2 η + 21 δ ) d ε ( x, y ) -ne ar a p ath-typ e c onfigur ation, i.e., ( x, y , z , w ) is (2 η + 21 δ ) d ε ( x, y ) -ne ar a typ e I I c onfigur ation. Pr o of. Let ( x ′ , y ′ , z ′ ) b e a path-t yp e configuration which is η d ε ( x, y )-near ( x, y , z ). By T h eorem 6.4, either ( x, y , w ) or ( w , y , x ) must b e 3 δ d ε ( x, z ) 6 6 1 − δ d ε ( x, y ) 6 7 δ d ε ( x, y )-near either a path-t yp e configuration or a tent- t yp e configuration. Supp ose first that ( x, y , w ) is 7 δd ε ( x, y )-near a tent- t yp e configuration ( x ′′ , y ′′ , w ′′ ). In this case, x ′′ is an ancesto r of y ′′ and h ( y ′′ ) − h ( x ′′ ) = d ε ( x ′′ , y ′′ ) > (1 − 14 δ ) d ε ( x, y ). At the same time, y ′ is an ancestor of x ′ and h ( x ′ ) − h ( y ′ ) = d ε ( x ′ , y ′ ) > (1 − 2 η ) d ε ( x, y ). So, 2( η + 7 δ ) d ε ( x, y ) > d ε ( y ′′ , y ′ ) + d ε ( x ′ , x ′′ ) > h ( y ′′ ) − h ( x ′′ ) + h ( x ′ ) − h ( y ′ ) > 2(1 − η − 7 δ ) d ε ( x, y ) , whic h is a con tradiction since η + 7 δ < 1 / 2. Next supp ose that ( w , y , x ) is 7 δ d ε ( x, y )-near a path-t yp e configuration ( w ′′ , y ′′ , x ′′ ). T hen | h ( x ′ ) − h ( x ′′ ) | 6 d ε ( x ′ , x ′′ ) 6 ( η + 7 δ ) d ε ( x, y ). So, ( η + 7 δ ) d ε ( x, y ) > d ε ( y ′ , y ′′ ) > h ( y ′′ ) − h ( y ′ ) = ( h ( y ′′ ) − h ( x ′′ )) + ( h ( x ′′ ) − h ( x ′ )) + h ( x ′ ) − h ( y ′ ) > 0 − ( η + 7 δ ) d ε ( x, y ) + (1 − 2 η ) d ε ( x, y ) , whic h is a con tradiction Lastly , sup p ose that ( w , y , x ) is 7 δ d ε ( x, y )-near a ten t-t yp e configuration ( w ′′ , y ′′ , x ′′ ). Note that | h ( y ′ ) − h ( y ′′ ) | 6 d ε ( y ′ , y ′′ ) 6 ( η + 7 δ ) d ε ( x, y ). S o, h ( y ′ ) > h ( y ′′ ) − ( η + 7 δ ) d ε ( x, y ). Also, h ( y ′′ ) − h ( w ′′ ) = d ε ( y ′′ , w ′′ ) > d ε ( y , w ) − 14 δ d ε ( x, y ) >  1 − δ 1 + δ − 14 δ  d ε ( x, y ) > ( η + 7 δ ) d ε ( x, y ) . 35 Consider the p oin t ¯ w defined as the ancestor of y ′ at distance h ( y ′′ ) − h ( w ′′ ) − ( η + 7 δ ) d ε ( x, y ) fr om y ′ . Let also w ′′′ b e the ancestor of y ′′ at d istance h ( y ′′ ) − h ( w ′′ ) − ( η + 7 δ ) d ε ( x, y ) fr om y ′′ . By Lemma 6.9, w e ha v e d ε ( ¯ w , w ′′′ ) 6 d ε ( y ′ , y ′′ ) 6 ( η + 7 δ ) d ε ( x, y ). Therefore, d ε ( ¯ w , w ) 6 d ε ( ¯ w , w ′′′ ) + d ε ( w ′′′ , w ′′ ) + d ε ( w ′′ , w ) 6 (2 η + 21 δ ) d ε ( x, y ) . Hence ( x, y , w ) is (2 η + 21 δ ) d ε ( x, y )-near the path-t yp e configuration ( x ′ , y ′ , ¯ w ). Lemma 6.19. L et ( x, y , z , w ) b e a δ -fork of B ∞ . Assume that ( x, y , z ) is η d ε ( x, y ) -ne ar a tent- typ e c onfigur ation. Assume also that η < 1 / 10 and ε n < 1 / 4 for al l n . Then ( w, y , x ) c annot b e η d ε ( x, y ) -ne ar a tent-typ e c onfigur ation. Pr o of. Let ( x ′ , y ′ , z ′ ) b e a tent t yp e confi guration that is η d ε ( x, y )-near ( x, y , z ). S u pp ose for con tra- diction that there exists a ten t t yp e configur ation ( w ′′ , y ′′ , x ′′ ) th at is η d ε ( x, y )-near ( w , y , x ). Note that h ( y ′′ ) > h ( y ′ ) − d ε ( y ′ , y ′′ ) > h ( y ′ ) − 2 η d ε ( x, y ) and h ( y ′ ) − h ( x ′ ) > (1 − 2 η ) d ε ( x, y ) > 2 η d ε ( x, y ). Let x ∗ b e the ancestor of y ′ at distance h ( y ′ ) − h ( x ′ ) − 2 η d ε ( x, y ) fr om y ′ , and let ˜ x b e the an- cestor of y ′′ at distance h ( y ′ ) − h ( x ′ ) − 2 η d ε ( x, y ) fr om y ′′ . An application of Lemma 6.9 yields the estimate d ε ( ˜ x, x ∗ ) 6 d ε ( y ′ , y ′′ ) 6 2 ηd ε ( x, y ). But, since h ( x ′′ ) > h ( y ′′ ), w e also kn o w that d ε ( ˜ x, x ′′ ) > h ( y ′′ ) − h ( ˜ x ) = d ε ( y ′ , x ′ ) − 2 η d ε ( x, y ). Hence, 2 η d ε ( x, y ) > d ε ( ˜ x, x ∗ ) > d ε ( ˜ x, x ′′ ) − d ε ( x ∗ , x ′ ) − d ε ( x ′ , x ′′ ) > d ε ( x ′ , y ′ ) − 6 η d ε ( x, y ) > (1 − 8 η ) d ε ( x, y ) , whic h is a con tradiction, since η < 1 / 10. Pr o of of L emma 6.14. Since ( x, y , z , w ) is a δ -fork, by Theorem 6.4, b oth ( x, y , z ) and ( x, y , w ) are 7 δ d ε ( x, y )-near a tent-t yp e configuration, a p ath-t yp e configuration, or th e corresp ond ing reve rse configurations. W e ha v e 10 p ossible com b inations of these p airs, a s app earing in T able 1. By applying Lemmas 6.18 and 6.19 with η = 7 δ , we ru le out three of these configur ations, and a fourth configuration is p ossible b ut only as 35 δ d ε ( x, y )-near a t yp e I I configuration. W e are left with six p ossible configurations. By applyin g Lemmas 6.16 and 6.17 w ith η = 7 δ w e conclude th at in t w o of those configurations w e hav e d ε ( w, z ) 6 (69 δ + 2 ε h 0 ) d ε ( x, y ), and the rest are configurations that are 7 δ d ε ( x, y )-near one of the t yp es I – I V . 6.2.3 Classification of a pproximate 3-pa t hs W e start with the follo wing natural notion: Definition 6.20. F or x 0 , x 1 , x 2 , x 3 ∈ B ∞ the qu adruple ( x 0 , x 1 , x 2 , x 3 ) i s c al le d a (1+ δ ) - appr oximate P 3 if ther e exists L > 0 such that for every 0 6 i 6 j 6 3 we have ( j − i ) L 6 d ε ( x i , x j ) 6 (1 + δ )( j − i ) L. Note that in this c ase x 1 ∈ Mid( x 0 , x 2 , δ ) and x 2 ∈ Mid( x 1 , x 3 , δ ) . As in the case of δ -forks, there are 10 p ossible concatenations of t w o midp oin ts configurations (path-t yp e or ten t-t yp e): P-P , P-p, P-T, P-t, p-P , p-T, p-t, T-T, T -t, t-T (the midp oin t config- urations p-p, P-p, t-p, T-p, p-P , t-P , T-P , t-t, T -t, t-T are resp ectiv ely su c h concatenati ons with the ord er of x 0 , x 1 , x 2 , x 3 rev ersed). W e will r ule out some of these p ossibilities, and obtain some stronger prop erties for the rest. S ee T able 2. As in the case of δ -forks, it will b e b eneficial to giv e names to thr ee sp ecial t yp es approximat e 3-paths: 36 Midp oint configuration Rev erse configuration T yp e (P-P) (p-p) t yp e A (P-p) (P-p) imp ossible (P-T) (t-p) imp ossible (P-t) (T-p) t yp e B (p-P) (p-P) imp ossible (p-T) (t-P) t yp e C (p-t) (T-P) impossib le (T-T) (t-t) p ossible only as t yp e C (T-t) (T-t) imp ossible (t-T) (t-T) imp ossible T able 2: The p ossible c onfigur ations of 3 p aths. Definition 6.21. F or x 0 , x 1 , x 2 , x 3 ∈ B ∞ and η > 0 , a quadruple ( x 0 , x 1 , x 2 , x 3 ) is c al le d: • η -ne ar a typ e A c onfigur ation if b oth ( x 0 , x 1 , x 2 ) and ( x 1 , x 2 , x 3 ) ar e η -ne ar p ath-typ e c onfig- ur ations, • η -ne ar a typ e B c onfigur ation if ( x 0 , x 1 , x 2 ) i s η -ne ar a p ath-typ e c onfigur ation, and ( x 3 , x 2 , x 1 ) is η - ne ar tent-typ e c onfigur ation, • η ne ar typ e C c onfigur ation if ( x 2 , x 1 , x 0 ) is η -ne ar a p ath-typ e c onfigur ation, and ( x 1 , x 2 , x 3 ) is η - ne ar a tent-typ e c onfigur ation. Se e also Figur e 6. T yp e C T yp e B x 0 x 1 x 2 x 3 x 0 x 1 x 2 x 3 T yp e A x 1 x 2 x 3 x 0 Figure 6: The thr e e p ossible typ es of appr oximate 3-p aths. The follo wing lemma is the main result of this sub s ection. Lemma 6.2 2. Assume that ε n < 1 4 for al l n and fix δ < 1 / 200 . A ssume that ( x 0 , x 1 , x 2 , x 3 ) is a (1 + δ ) -appr oximate P 3 . Then either ( x 0 , x 1 , x 2 , x 3 ) or ( x 3 , x 2 , x 1 , x 0 ) is 35 δ d ε ( x 0 , x 1 ) -ne ar a c onfigur ation of typ e A , B or C . 37 The pro of of Lemma 6.22 is again a case analysis th at examines all 10 p ossible wa ys (up to symmetry) to concatenate tw o midp oin t configur ations. The p ro of is divided into a few lemmas according to the cases, and is completed at th e end of this subsection. Lemma 6.23. Assume that ε n < 1 4 for al l n and that ( x 0 , x 1 , x 2 , x 3 ) is a (1 + δ ) -appr oximate P 3 such that ( x 0 , x 1 , x 2 ) is η d ε ( x 0 , x 1 ) -ne ar a p ath-typ e c onfigur ation. If max { δ , η } < 1 / 200 then e ither ( x 1 , x 2 , x 3 ) is 7 δ d ε ( x 0 , x 1 ) -ne ar a p ath-typ e c onfigur ation (typ e A ), or ( x 3 , x 2 , x 1 ) is 7 δ d ε ( x 0 , x 1 ) - ne ar a tent-typ e c onfigur ation (typ e B ). Pr o of. Due to Th eorem 6.4 we only need to ru le out the p ossibilit y that ( x 3 , x 2 , x 1 ) is 7 δ d ε ( x 1 , x 2 )- near a p ath-t yp e confi guration, or that ( x 1 , x 2 , x 3 ) is 7 δ d ε ( x 1 , x 2 )-near a ten t-type confi gu r ation. Let ( x ′ 0 , x ′ 1 , x ′ 2 ) b e a path-t yp e configur ation that is η d ε ( x 0 , x 1 )-near ( x 0 , x 1 , x 2 ). Supp ose first that ( x 3 , x 2 , x 1 ) is 7 δ d ε ( x 1 , x 2 )-near th e p ath-t yp e configur ation ( x ′′ 3 , x ′′ 2 , x ′′ 1 ). S ince h ( x ′ 1 ) > h ( x ′ 2 ) and h ( x ′′ 2 ) > h ( x ′′ 1 ) we ha v e, | h ( x 1 ) − h ( x 2 ) | 6 | h ( x 1 ) − h ( x ′ 1 ) | +  h ( x ′ 1 ) − h ( x ′ 2 )  + | h ( x ′ 2 ) − h ( x 2 ) | 6 d ε ( x 1 , x ′ 1 ) +  h ( x ′ 1 ) − h ( x ′ 2 )  + d ε ( x ′ 2 , x 2 ) 6 2 η d ε ( x 0 , x 1 ) +  h ( x ′ 1 ) − h ( x ′ 2 )  , (74 ) and similarly , | h ( x 1 ) − h ( x 2 ) | 6 | h ( x 1 ) − h ( x ′′ 1 ) | +  h ( x ′′ 2 ) − h ( x ′′ 1 )  + | h ( x ′′ 2 ) − h ( x 2 ) | 6 d ε ( x 1 , x ′′ 1 ) +  h ( x ′′ 2 ) − h ( x ′′ 1 )  + d ε ( x ′ 2 , x 2 ) 6 14 δd ε ( x 0 , x 1 ) +  h ( x ′′ 2 ) − h ( x ′′ 1 )  . (75 ) By su mming (74) and (75) w e obtain the b ound 2 | h ( x 1 ) − h ( x 2 ) | 6 (2 η + 14 δ ) d ε ( x 0 , x 1 ) + d ε ( x ′ 1 , x ′′ 1 ) + d ε ( x ′ 2 , x ′′ 2 ) 6 (4 η + 28 δ ) d ε ( x 0 , x 1 ) . Th us | h ( x 1 ) − h ( x 2 ) | 6 (2 η + 14 δ ) d ε ( x 0 , x 1 ) . ( 76) Since x ′ 0 is a descendan t of x ′ 1 , | h ( x 0 ) − h ( x 1 ) − d ε ( x 0 , x 1 ) | 6 | h ( x ′ 0 ) − h ( x ′ 1 ) − d ε ( x 0 , x 1 ) | + 2 η d ε ( x 0 , x 1 ) = | d ε ( x ′ 0 , x ′ 1 ) − d ε ( x 0 , x 1 ) | + 2 δ d ε ( x 0 , x 1 ) 6 4 η d ε ( x 0 , x 1 ) . (77 ) Similarly , since x ′′ 3 is a descendan t of x ′′ 2 , | h ( x 3 ) − h ( x 2 ) − d ε ( x 0 , x 1 ) | 6 | h ( x ′′ 3 ) − h ( x ′′ 2 ) − d ε ( x 0 , x 1 ) | + 14 δ d ε ( x 1 , x 2 ) = | d ε ( x ′′ 3 , x ′′ 2 ) − d ε ( x 0 , x 1 ) | + 14 δd ε ( x 0 , x 1 ) 6 28 δ d ε ( x 0 , x 1 ) . (78 ) Hence, | h ( x ′′ 3 ) − h ( x ′ 0 ) | 6 | h ( x ′′ 3 ) − h ( x 3 ) | + | h ( x 3 ) − h ( x 2 ) − d ε ( x 0 , x 1 ) | + | h ( x 2 ) − h ( x 1 ) | + | h ( x 0 ) − h ( x 1 ) − d ε ( x 0 , x 1 ) | + | h ( x 0 ) − h ( x ′ 0 ) | (76) ∧ (77) ∧ (78) 6 d ε ( x ′′ 3 , x 3 ) + 28 δ d ε ( x 0 , x 1 ) + (2 η + 14 δ ) d ε ( x 0 , x 1 ) + 4 η d ε ( x 0 , x 1 ) + d ε ( x 0 , x ′ 0 ) 6 (49 δ + 7 η ) d ε ( x 0 , x 1 ) . (79) 38 W e record for f u ture r eference the follo wing consequence of (76) and (79): min { h ( x ′ 0 ) , h ( x ′′ 3 ) } − min { h ( x ′ 1 ) , h ( x ′′ 2 ) } 6 m ax  h ( x ′ 0 ) − h ( x ′ 1 ) , h ( x ′ 0 ) − h ( x ′′ 2 )  (79) 6 max  d ε ( x ′ 0 , x ′ 1 ) , h ( x ′′ 3 ) − h ( x ′′ 2 ) + (49 δ + 7 η ) d ε ( x 0 , x 1 )  6 max  (1 + 2 η ) d ε ( x 0 , x 1 ) , d ε ( x ′′ 3 , x ′′ 2 ) + (49 δ + 7 η ) d ε ( x 0 , x 1 )  6 (1 + 64 δ + 7 η ) d ε ( x 0 , x 1 ) . (80) W e next claim that lca ( x ′ 0 , x ′′ 3 ) = lca ( x ′ 1 , x ′′ 2 ) . (81) Indeed, since x ′ 1 is an ancestor of x ′ 0 and x ′′ 2 is an ancestor of x ′′ 3 , if lca ( x ′ 0 , x ′′ 3 ) 6 = lca ( x ′ 1 , x ′′ 2 ) then either x ′ 1 is a descendan t of x ′′ 2 , or x ′′ 2 is a descendan t of x ′ 1 . If x ′ 1 is a descendan t of x ′′ 2 then ( η + 7 δ ) d ε ( x 0 , x 1 ) > d ε ( x ′ 1 , x ′′ 1 ) > d ε ( x ′′ 2 , x ′ 1 ) > d ε ( x 2 , x 1 ) − ( η + 7 δ ) d ε ( x 0 , x 1 ) > 1 1 + δ d ε ( x 0 , x 1 ) − ( η + 7 δ ) d ε ( x 0 , x 1 ) , whic h is a con tradiction since δ, η < 1 / 200. S imilarly , if x ′′ 2 is a descendant of x ′ 1 then ( η + 7 δ ) d ε ( x 0 , x 1 ) > d ε ( x ′ 2 , x ′′ 2 ) > d ε ( x ′′ 2 , x ′ 1 ) > 1 1 + δ d ε ( x 0 , x 1 ) − ( η + 7 δ ) d ε ( x 0 , x 1 ) , arriving once more at a con tradiction. This p ro v es (81) . No w, 3 1 + δ d ε ( x 0 , x 1 ) 6 d ε ( x 0 , x 3 ) 6 d ε ( x ′′ 3 , x ′ 0 ) + ( η + 7 δ ) d ε ( x 0 , x 1 ) (79) 6 2 ε min { h ( x ′ 0 ) ,h ( x ′′ 3 ) }  min { h ( x ′ 0 ) , h ( x ′′ 3 ) } − h ( lca ( x ′ 0 , x ′′ 3 ))  + (8 η + 56 δ ) d ε ( x 0 , x 1 ) (81) = 2 ε min { h ( x ′ 0 ) ,h ( x ′′ 3 ) }  min { h ( x ′ 1 ) , h ( x ′′ 2 ) } − h ( lca ( x ′ 1 , x ′′ 2 ))  + (8 η + 56 δ ) d ε ( x 0 , x 1 ) + 2 ε min { h ( x ′ 0 ) ,h ( x ′′ 3 ) }  min { h ( x ′ 0 ) , h ( x ′′ 3 ) } − min { h ( x ′ 1 ) , h ( x ′′ 2 ) }  (80) 6 2 ε min { h ( x ′ 1 ) ,h ( x ′′ 2 ) }  min { h ( x ′ 1 ) , h ( x ′′ 2 ) } − h ( lca ( x ′ 1 , x ′′ 2 ))  +  8 η + 56 δ + 1 + 64 δ + 7 η 2  d ε ( x 0 , x 1 ) (82) 6 d ε ( x ′ 1 , x ′′ 2 ) +  1 2 + 88 δ + 12 η  d ε ( x 0 , x 1 ) 6  3 2 + 96 δ + 13 η  d ε ( x 0 , x 1 ) , (83) where in (82) we used min { h ( x ′ 0 ) , h ( x ′′ 3 ) } > min { h ( x ′ 1 ) , h ( x ′′ 2 ) } and ε min { h ( x ′ 0 ) ,h ( x ′′ 3 ) } < 1 / 4. Since max { η , δ } < 1 / 200, the b ound (83) is a con tradiction. 39 Next supp ose th at ( x 1 , x 2 , x 3 ) is 7 δ d ε ( x 0 , x 1 )-near a ten t-t yp e configur ation ( x ′′ 1 , x ′′ 2 , x ′′ 3 ). Since h ( x ′ 2 ) 6 h ( x ′ 1 ) and h ( x ′′ 2 ) > h ( x ′′ 1 ), we ha v e | h ( x 1 ) − h ( x 2 ) | 6 | h ( x 1 ) − h ( x ′ 1 ) | +  h ( x ′ 1 ) − h ( x ′ 2 )  + | h ( x ′ 2 ) − h ( x 2 ) | 6 d ε ( x 1 , x ′ 1 ) +  h ( x ′ 1 ) − h ( x ′ 2 )  +  h ( x ′′ 2 ) − h ( x ′′ 1 )  + d ε ( x ′ 2 , x 2 ) 6 d ε ( x 1 , x ′ 1 ) + d ε ( x ′ 1 , x ′′ 1 ) + d ε ( x ′ 2 , x ′′ 2 ) + d ε ( x ′ 2 , x 2 ) 6 (4 η + 14 δ ) d ε ( x 0 , x 1 ) . (84) On th e other hand, x ′′ 1 is an ancestor of x ′′ 2 , and therefore we ha v e  1 1 + δ − 14 δ  d ε ( x 0 , x 1 ) 6 d ε ( x ′′ 1 , x ′′ 2 ) = h ( x ′′ 2 ) − h ( x ′′ 1 ) 6 | h ( x 1 ) − h ( x 2 ) | + 14 δ d ε ( x 0 , x 1 ) (84) 6 (4 η + 28 δ ) d ε ( x 0 , x 1 ) , (85 ) whic h is a con tradiction since max { η , δ } < 1 / 200 . Lemma 6.24. Assume that ε n < 1 4 for al l n and that ( x 0 , x 1 , x 2 , x 3 ) is a (1 + δ ) -appr oximat e P 3 such that ( x 2 , x 1 , x 0 ) is η d ε ( x 0 , x 1 ) -ne ar a p ath-typ e c onfigur ation. If m ax { δ, η } < 1 / 200 then either ( x 3 , x 2 , x 1 ) is 7 δ d ε ( x 0 , x 1 ) -ne ar a p ath-typ e c onfigur ation (r everse typ e A ), or ( x 1 , x 2 , x 3 ) is 7 δ d ε ( x 0 , x 1 ) -ne ar a tent-typ e c onfigur ation (typ e C ). Pr o of. Let ( x ′ 2 , x ′ 1 , x ′ 0 ) b e in pat h-t yp e configuration th at is η d ε ( x 0 , x 1 )-near ( x 2 , x 1 , x 0 ). First, assume for con tradiction that ( x 3 , x 2 , x 1 ) is 7 δ d ε ( x 1 , x 2 )-near a tent- t yp e configuration ( x ′′ 3 , x ′′ 2 , x ′′ 1 ). Then h ( x ′′ 1 ) > h ( x ′′ 2 ), where as h ( x ′ 2 ) − h ( x ′ 1 ) = d ε ( x ′ 2 , x ′ 1 ). Arguing as in (84) , it follo ws that | h ( x 1 ) − h ( x 2 ) | 6 (2 η + 28 δ ) d ε ( x 0 , x 1 ), and we arriv e at a contradict ion by arguing similarly to (85). Next, assu m e for con tradiction that ( x 1 , x 2 , x 3 ) is 7 δ d ε ( x 1 , x 2 )-near a path-type configuration ( x ′′ 1 , x ′′ 2 , x ′′ 3 ). T h en h ( x ′′ 1 ) − h ( x ′′ 2 ) = d ε ( x ′′ 1 , x ′′ 2 ), w hereas h ( x ′ 2 ) − h ( x ′ 1 ) = d ε ( x ′ 1 , x ′ 2 ). By summ in g these tw o iden tities, w e arriv e at a cont radiction as follo ws :  2 1 + δ − 2 η − 14 δ  d ε ( x 0 , x 1 ) 6 d ε ( x ′ 1 , x ′ 2 ) + d ε ( x ′′ 1 , x ′′ 2 ) =  h ( x ′ 2 ) − h ( x ′′ 2 )  +  h ( x ′′ 1 ) − h ( x ′ 1 )  6 d ε ( x ′ 2 , x ′′ 2 ) + d ε ( x ′ 1 , x ′′ 1 ) 6 (2 η + 14 δ ) d ε ( x 0 , x 1 ) . Lemma 6.25. Assume that ε n < 1 4 for al l n and that ( x 0 , x 1 , x 2 , x 3 ) is a (1 + δ ) -appr oximate P 3 such that ( x 0 , x 1 , x 2 ) is η d ε ( x 0 , x 1 ) -ne ar a tent-typ e c onfigur ation. If max { δ , η } < 1 / 200 then either ( x 2 , x 1 , x 0 ) is (14 δ + 3 η ) d ε ( x 0 , x 1 ) -ne ar a p ath-typ e c onfigur ation and ( x 1 , x 2 , x 3 ) is 7 δ d ε ( x 1 , x 2 ) - ne ar a tent-typ e c onfigur ation (typ e C ), or ( x 3 , x 2 , x 1 ) is 7 δ d ε ( x 0 , x 1 ) -ne ar a p ath-typ e c onfigur ation (r everse typ e B ). Pr o of. Let ( x ′ 0 , x ′ 1 , x ′ 2 ) b e a tent-t yp e configuration that is η d ε ( x 0 , x 1 )-near ( x 0 , x 1 , x 2 ). First, supp ose that ( x 1 , x 2 , x 3 ) is 7 δd ε ( x 1 , x 2 )-near a ten t-t yp e configuration ( x ′′ 1 , x ′′ 2 , x ′′ 3 ). Not e that | h ( x ′ 1 ) − h ( x ′′ 1 ) | 6 d ε ( x ′ 1 , x ′′ 1 ) 6 ( η + 7 δ ) d ε ( x 0 , x 1 ). So, let x ′′ 0 b e an ancestor of x ′′ 1 at d istance h ( x ′ 1 ) − h ( x ′ 0 ) − ( η + 7 δ ) d ε ( x 0 , x 1 ) ∈ [0 , h ( x ′′ 1 )] from x ′′ 1 , and let x ∗ 0 b e an ancestor of x ′ 1 at dis- tance h ( x ′ 1 ) − h ( x ′ 0 ) − ( η + 7 δ ) d ε ( x 0 , x 1 ) from x ′ 1 . Th en h ( x ′ 1 ) − h ( x ∗ 0 ) = h ( x ′′ 1 ) − h ( x ′′ 0 ) and d ε ( x ∗ 0 , x ′ 0 ) 6 ( η + 7 δ ) d ε ( x 0 , x 1 ). By Lemma 6.9, d ε ( x 0 , x ′′ 0 ) − (2 η + 7 δ ) d ε ( x 0 , x 1 6 d ε ( x 0 , x ′′ 0 ) − d ε ( x ∗ 0 , x ′ 0 ) − d ε ( x ′ 0 , x 0 ) 6 d ε ( x ∗ 0 , x ′′ 0 ) 6 d ε ( x ′ 1 , x ′′ 1 ) 6 ( η + 7 δ ) d ε ( x 0 , x 1 ) . 40 Hence ( x ′′ 2 , x ′′ 1 , x ′′ 0 ) is a path-t yp e configuration that is (14 δ + 3 η ) d ε ( x 0 , x 1 )-near ( x 2 , x 1 , x 0 ). Next assume for con trad iction that ( x 3 , x 2 , x 1 ) is 7 δ d ε ( x 1 , x 2 ) near a ten t-type configuration ( x ′′ 3 , x ′′ 2 , x ′′ 1 ). Th en (1 − 15 δ ) d ε ( x 0 , x 1 ) 6  1 1 + δ − 14 δ  d ε ( x 0 , x 1 ) 6 h ( x ′′ 2 ) − h ( x ′′ 3 ) 6 (1 + 15 δ ) d ε ( x 0 , x 1 ) , (86) and (1 − δ − 2 η ) d ε ( x 0 , x 1 ) 6  1 1 + δ − 2 η  d ε ( x 0 , x 1 ) 6 h ( x ′ 1 ) − h ( x ′ 0 ) 6 (1 + δ + 2 η ) d ε ( x 0 , x 1 ) . (87) So, let x ## 3 b e an ancestor of x ′′ 2 at distance h ( x ′′ 2 ) − h ( x ′′ 3 ) − (16 δ + 2 η ) d ε ( x 0 , x 1 ) ∈ [0 , h ( x ′′ 2 )] fr om x ′′ 2 , and let x # 0 b e an ancestor of x ′ 1 at d istance h ( x ′′ 2 ) − h ( x ′′ 3 ) − (16 δ + 2 η ) d ε ( x 0 , x 1 ) ∈ [0 , h ( x ′ 1 )] from x ′ 1 . Th en d ε ( x ′′ 3 , x ## 3 ) 6 (16 δ + 2 η ) d ε ( x 0 , x 1 ) , (88) and d ε ( x ′ 0 , x # 0 ) =    h ( x ′ 0 ) − h ( x # 0 )    =   h ( x ′ 0 ) −  h ( x ′ 1 ) − h ( x ′′ 2 ) + h ( x ′′ 3 ) + (16 δ + 2 η ) d ε ( x 0 , x 1 )    (86) ∧ (87) 6 2(1 6 δ + 2 η ) d ε ( x 0 , x 1 ) . (89 ) Moreo ve r, h ( x 1 ) − h ( x # 0 ) = h ( x ′′ 2 ) − h ( x ## 3 ), so b y Lemma 6.9 w e ha ve  3 1 + δ − 55 δ − 7 η  d ε ( x 0 , x 1 ) 6 d ε ( x 0 , x 3 ) − (55 δ + 7 η ) d ε ( x 0 , x 1 ) (88) ∧ (89) 6 d ε ( x # 0 , x ## 3 ) 6 d ε ( x ′ 1 , x ′′ 2 ) 6 (1 + 8 δ + η ) d ε ( x 0 , x 1 ) , whic h is a con tradiction since max { δ, η } < 1 / 200. Lastly , assume for contradictio n that ( x 1 , x 2 , x 3 ) is 7 δd ε ( x 1 , x 2 )-near a path-t yp e configuration ( x ′′ 1 , x ′′ 2 , x ′′ 3 ). Th en since h ( x ′ 1 ) 6 h ( x ′ 2 ) we ha v e  1 1 + δ − 14 δ  d ε ( x 0 , x 1 ) 6 d ε ( x ′′ 1 , x ′′ 2 ) = h ( x ′′ 1 ) − h ( x ′′ 2 ) 6  h ( x ′′ 1 ) − h ( x ′′ 2 )  +  h ( x ′ 2 ) − h ( x ′ 1 )  6 d ε ( x ′′ 1 , x ′ 1 ) + d ε ( x ′′ 2 , x ′ 2 ) 6 (14 δ + 2 η ) d ε ( x 0 , x 1 ) , a contradict ion. Lemma 6.26. Assume that ε n < 1 4 for al l n and that ( x 0 , x 1 , x 2 , x 3 ) is a (1 + δ ) -appr oximat e P 3 such that ( x 2 , x 1 , x 0 ) is η d ε ( x 0 , x 1 ) -ne ar a tent- typ e c onfigur ation. If max { δ, η } < 1 / 200 then ( x 1 , x 2 , x 3 ) c annot b e 7 δ d ε ( x 0 , x 1 ) ne ar a tent-typ e c onfigur ation. Pr o of. Let ( x ′ 2 , x ′ 1 , x ′ 0 ) b e a tent- t yp e configuration that is η d ε ( x 0 , x 1 )-near ( x 2 , x 1 , x 0 ). Su p p ose for con tradiction that ( x 1 , x 2 , x 3 ) is 7 δ d ε ( x 0 , x 1 )-near a ten t-t yp e configuration ( x ′′ 1 , x ′′ 2 , x ′′ 3 ). Then h ( x ′ 1 ) − h ( x ′ 2 ) = d ε ( x ′ 1 , x ′ 2 ), whereas h ( x ′′ 2 ) − h ( x ′′ 1 ) = d ε ( x ′′ 1 , x ′′ 2 ). T aking the s u m of these tw o inequalities w e conclude that d ε ( x ′′ 1 , x ′′ 2 ) + d ε ( x ′ 1 , x ′ 2 ) 6 d ε ( x ′ 1 , x ′′ 1 ) + d ε ( x ′ 2 , x ′′ 2 ) 6 (2 η + 14 δ ) d ε ( x 0 , x 1 ) . A t the same time,  2 1+ δ − 2 η − 14 δ  d ε ( x 0 , x 1 ) 6 d ε ( x ′′ 1 , x ′′ 2 ) + d ε ( x ′ 1 , x ′ 2 ), w h ic h leads to the desired con tradiction. 41 Pr o of of L emma 6.22. Since ( x 0 , x 1 , x 2 , x 3 ) is a (1 + δ )-approximat e P 3 , w e ha v e x 1 ∈ Mid( x 0 , x 2 , δ ), and x 2 ∈ Mid( x 1 , x 3 , δ ). S ince the assu m ptions of Th eorem 6.4 hold, w e can app ly with η = 7 δ Lemmas 6.23, 6.24, 6.25, 6.26, and conclud e that either ( x 0 , x 1 , x 2 , x 3 ) or ( x 3 , x 2 , x 1 , x 0 ) m ust b e 35 δd ε ( x 0 , x 1 )-near a configuration of t yp e A , B or C . 6.3 Nonem b eddabilit y of vertically faithful B 4 In wh at follo w s we need some standard notation on trees. As b efore, B n is the complete binary tree of h eigh t n ; the ro ot of B n is denoted by r . Denote by I ( B n ) the s et of internal v ertices of B n , i.e., v ertices of B n whic h are not the ro ot or a leaf. F or a v er tex v in { r } ∪ I ( B n ) we denote b y v 0 and v 1 its children. F or α ∈ { 0 , 1 } ∗ (the set of finite sequ ences of ’0’ and ’1’) and a ∈ { 0 , 1 } w e denote by v αa = ( v α ) a . The aim of the curren t section is to pro ve the follo wing lemma. Lemma 6.27. Fix 0 < δ < 1 / 400 and let f : B 4 → ( B ∞ , d ε ) b e a (1 + δ ) -ve rtic al ly faithful emb e dding. Then the distortion of f satisfies dist( f ) > 1 500 δ + ε h 0 , wher e h 0 = min x ∈ B 4 h ( f ( x )) . The pro of of Lemma 6.27 is by a con tradiction. By L emma 6.14, assumin g the d istortion of f is small, all the δ -forks in the (1 + δ )-v ertically faithful em b ed d ing m ust b e of t yp es I – I V . By exploring the constrains implied by Lemma 6.22 on ho w those δ -forks can b e “stit c hed” together, w e r eac h the conclusion th at they are su ffi cien tly sev ere to force an y v ertically faithful em b edding of B 4 to hav e a large con traction, and therefore high distortion. Fix f : B 4 → ( B ∞ , d ε ). F or u ∈ I ( B 4 ) w e denote by F ( u ) the fork in wh ic h u is the center p oint, i.e., if v b e the parent of u in B 4 , then F ( u ) def = ( f ( v ) , f ( u ) , f ( u 0 ) , f ( u 1 )) . W e shall assume from no w on th at f satisfies the assumptions of Lemm a 6.27, i.e., that it satisfies (8) with D = 1 + δ for some δ < 1 / 400 an d λ > 0. Lemma 6.28. Fix u ∈ B 4 with h ( u ) ∈ { 1 , 2 } . If the fork F ( u ) is 37 δ λ -ne ar a typ e I or typ e I I I c onfigur ation, then ther e exists w ∈ I ( B 4 ) satisfying d ε ( f ( w 0 ) , f ( w 1 )) 6 (170 δ + ε h 0 ) · 2 λ. (90) Pr o of. Let v b e the p aren t of u . Hence, ( f ( v ) , f ( u ) , f ( u 0 ) , f ( u 1 )) is 35 δ (1 + δ ) λ -near a t yp e I or a t yp e I I I configu r ation. Assu me first that ( f ( v ) , f ( u ) , f ( u 0 ) , f ( u 1 )) is 37 δλ -near a t y p e I configu- ration. If b oth ( f ( u 0 ) , f ( u ) , f ( v )) and ( f ( u 1 ) , f ( u ) , f ( v )) w ere 37 δλ -n ear a path t yp e configur ation then by Lemma 6.16 (with η = 37 δ ) we w ould h a ve d ε ( f ( u 0 ) , f ( u 1 )) 6 (339 δ + 2 ε h 0 )(1 + δ ) λ 6 (170 δ + ε h 0 ) · 2 λ, (91) pro ving (90) with w = u . The same conclusion holds when ( f ( v ) , f ( u ) , f ( u 0 ) , f ( u 1 )) is 37 δ λ -near a t y p e I I I co nfigur ation: in this case without loss of generalit y ( f ( v ) , f ( u ) , f ( u 0 )) is 37 δλ -near 42 a ten t-t yp e confi guration and ( f ( u 1 ) , f ( u ) , f ( v )) is 37 δλ -near a p ath-t yp e confi gu r ation. Using Lemma 6.16 as ab ov e we would arriv e at the conclusion (91) if ( f ( u 0 ) , f ( u ) , f ( v )) were 37 δ λ -near a path-t yp e t yp e configur ation. Thus, in b oth the t yp e I and t yp e I I I cases of Lemma 6.28 we ma y assume that ( f ( v ) , f ( u ) , f ( u 0 )) is 37 δλ -near a ten t-t yp e configuration, and that, b y Lemma 6.8, ( f ( v ) , f ( u ) , f ( u 0 )) is not 37 δ λ -near a path-type configuration, and ( f ( u 0 ) , f ( u ) , f ( v )) is not 37 δλ - near a path-t yp e configuration or a ten t-t yp e configuration. By Lemma 6.22 (and T ab le 2) ( f ( u 0 c ) , f ( u 0 ) , f ( u ) , f ( v )) m u st b e 35 δ (1 + δ ) λ -near a typ e B configuration for b oth c ∈ { 0 , 1 } . This means that ( f ( u 0 c ) , f ( u 0 ) , f ( u )) are b oth 35 δ (1 + δ ) λ - near a path-type configur ation, and so b y Lemma 6.16 (with η = 35 δ (1 + δ )) w e d educe th at d ε ( f ( u 00 ) , f ( u 01 )) 6 (170 δ + ε h 0 ) · 2 λ . Lemma 6.29. Fix u ∈ B 4 with h ( u ) ∈ { 1 , 2 } . If F ( u ) is 37 δ λ -ne ar a typ e I I c onfigur ation then for b oth b ∈ { 0 , 1 } either F ( u b ) i s 99 δ λ -ne ar a typ e I I c onfigur ation, or d ε ( f ( u b 0 ) , f ( u b 1 )) 6 400 δλ . Pr o of. Let v b e th e p aren t of u . F or both c ∈ { 0 , 1 } w e kno w that ( f ( v ) , f ( u ) , f ( u 0 ) , f ( u 0 c )) is a (1 + δ )-app r o xim ate P 3 , an d therefore b y Lemma 6.22 either ( f ( v ) , f ( u ) , f ( u 0 ) , f ( u 0 c )) or ( f ( u 0 c ) , f ( u 0 ) , f ( u ) , f ( v )) is 35 δ (1 + δ ) λ -near a configur ation of type A , B or C . Note that since ( f ( v ) , f ( u ) , f ( u 0 )) is assu med to b e 37 δ λ -near a path-t yp e configuration, we rule out the p ossib ilit y that ( f ( v ) , f ( u ) , f ( u 0 ) , f ( u 0 c )) is 35 δ (1 + δ ) λ -near a configuration of t yp e C , since otherwise b oth ( f ( v ) , f ( u ) , f ( u 0 )) and ( f ( u 0 ) , f ( u ) , f ( v )) would b e 37 δ λ -near path-t yp e configurations, con tra- dicting Lemma 6.8. F or the same r eason we rule out the p ossibilit y th at ( f ( u 0 c ) , f ( u 0 ) , f ( u ) , f ( v )) is 3 5 δ (1 + δ ) λ -near a c onfiguration of t yp e A or t yp e B . An insp ection o f the three remain- ing p ossibilities shows that either ( f ( u ) , f ( u 0 ) , f ( u 0 c )) is 37 δ λ -n ear a path-t yp e configuration, or ( f ( u 0 c ) , f ( u 0 ) , f ( u )) is 37 δ λ -near a ten t-t yp e configuration. No w, • If for b oth c ∈ { 0 , 1 } we ha v e that ( f ( u ) , f ( u 0 ) , f ( u 0 c )) is 37 δλ -near a path-t yp e configuration, then F ( u 0 ) is 37 δλ -near a t yp e I I configur ation. • If for b oth c ∈ { 0 , 1 } we ha v e that ( f ( u 0 c ) , f ( u 0 ) , f ( u )) are 37 δ λ -near a ten t-type configura- tion, then b y Lemma 6.17 w e ha ve d ε ( f ( u 01 ) , f ( u 00 )) 6 400 δ . • By Lemma 6.18, th e only wa y that ( f ( u ) , f ( u 0 ) , f ( u 00 )) could b e 37 δ λ -near a path t yp e con- figuration wh ile at the same time ( f ( u 01 ) , f ( u 0 ) , f ( u )) is 37 δ λ -near a tent- t yp e configuration (or vice v ersa), is th at F ( u 0 ) is 99 δλ -near a t yp e I I configur ation. Lemma 6.30. Fix u ∈ B 4 with h ( u ) ∈ { 1 , 2 } . If F ( u ) i s 35 δ (1 + δ ) λ -ne ar a typ e I V c onfigur ation, then ther e exists b ∈ { 0 , 1 } such that F ( u b ) is 37 δ λ -ne ar a typ e I I c onfigur ation. Pr o of. Let v b e the p arent of u . Without loss of generalit y ( f ( u 0 ) , f ( u ) , f ( v )) is 35 δ (1 + δ ) λ -near a ten t-type configur ation. By Lemma 6.22 (usin g Lemma 6.8 to ru le out the remaining p ossibilities), this means that for b oth c ∈ { 0 , 1 } the quadrup le ( f ( u 0 c ) , f ( u 0 ) , f ( u ) , f ( v )) is 35 δ (1 + δ ) 2 λ -near a t yp e C configuration, and therefore F ( u 0 ) is 35 δ (1 + δ ) 2 λ n ear a t yp e I I configur ation. Lemma 6.31. Fix u ∈ B 4 with h ( u ) ∈ { 0 , 1 , 2 } . If F ( u 0 ) and F ( u 1 ) ar e b oth 99 δ λ -ne ar typ e a I I c onfigur ation then d ε ( f ( u 0 ) , f ( u 1 )) 6 1000 δ λ. 43 Pr o of. By our assumptions, ( f ( u ) , f ( u 0 ) , f ( u 00 )) is 99 δλ -near a path typ e configur ation ( u ′ , u ′ 0 , u ′ 00 ) and ( f ( u ) , f ( u 1 ) , f ( u 10 )) is 99 δ λ -near a path-type configuration ( u ′′ , u ′′ 1 , u ′′ 10 ). W e ma y assume without loss of g eneralit y that h ( u ′′ ) − h ( u ′ 1 ) 6 h ( u ′ ) − h ( u ′ 0 ). W e ma y therefore consider the ancestor u ∗ 1 of u ′ suc h that h ( u ′ ) − h ( u ∗ 1 ) = h ( u ′′ ) − h ( u ′′ 1 ), implying in particular that h ( u ∗ 1 ) > h ( u ′ 0 ) (recall that u ′ 0 is an ancesto r of u ′ , and u ′′ 1 is ancestor of u ′′ ). By L emma 6.9 w e ha v e d ε ( u ∗ 1 , u ′′ 1 ) 6 d ε ( u ′ , u ′′ ) 6 198 δλ. (92) Hence, h ( u ′ ) − h ( u ∗ 1 ) = d ε ( u ′ , u ∗ 1 ) (92) > d ε ( u ′ , u ′′ 1 ) − 198 δλ > d ε ( f ( u ) , f ( u 1 )) − 394 δλ > (1 − 394 δ ) λ. (93) But, we also kn o w that h ( u ′ ) − h ( u ′ 0 ) = d ε ( u ′ , u ′ 0 ) 6 d ε ( f ( u ) , f ( u 0 )) + 198 δλ 6 (1 + 200 δ ) λ. (94) It follo w s from (93) and (94) that d ε ( u ′ 0 , u ∗ 1 ) = h ( u ∗ 1 ) − h ( u ′ 0 ) 6 601 δ λ . Th erefore, d ε ( f ( u 1 ) , f ( u 0 )) 6 d ε ( f ( u 0 ) , u ′ 0 ) + d ε ( u ′ 0 , u ∗ 1 ) + d ε ( u ∗ 1 , u ′′ 1 ) + d ε ( u ′′ 1 , f ( u 1 )) = 1000 δ λ. Pr o of of L emma 6.27. W e ma y assume that for all u ∈ I ( B 4 ) the fork F ( u ) is 35 δ (1 + δ ) λ -near a configuration of t yp e I , I I , I I I , or I V . In deed, otherwise the pro of is complete by Lemma 6.14. If F ( r 0 ) is 35 δ (1 + δ ) λ -near a t yp e I or t yp e I I I configuration, th en by Lemma 6.28 the p ro of is complete. If F ( r 0 ) is 35 δ (1 + δ ) λ -near a t yp e I V configuration then by Lemma 6.30 there exists b ∈ { 0 , 1 } su c h that F ( r 0 b ) is 37 δ λ -near a t yp e I I configuration. It therefore remains to deal with the case in whic h for some u ∈ { r 0 , r 0 b } the fork F ( u ) is 37 δ λ -n ear a t yp e I I configuration. Ap p lying Lemma 6.29, either w e are d one, or b oth F ( u 0 ) and F ( u 1 ) are 99 δ λ -near a t yp e I I configuration, but th en b y Lemma 6.31 the pro of of Lemma 6.27 is complete. 6.4 Nonem b eddabilit y of binary trees W e are no w in p osition to complete the pro of of Theorem 1.10. Pr o of of The or em 1.10. W rite ε n = 1 /s ( n ), a nd ε = { ε n } ∞ n =0 . Thus { ε n } ∞ n =0 is non-increasing, { nε n } ∞ n =0 is non -d ecreasing, and ε n 6 1 / 4. W e can therefore c h o ose the metric space ( X , d X ) = ( B ∞ , d ε ). Th e id en tit y em b edding of B n in to the top n -leve ls of B ∞ sho ws that c X ( B n ) 6 s ( n ). It remains to prov e the lo we r b ound on c X ( B n ). T o this en d take an arbitrary injectio n f : B n → X satisfying d ist( f ) 6 s ( n ), and w e will now p r o ve that dist( f ) > s  n 40 s ( n )   1 − C s ( n ) log s ( n ) log n  . (95) By adju sting the constan t C in (95), w e may assume b elo w that n is large enough, s a y , n > 100. W rite h 0 = ⌊ n/ (40 s ( n )) ⌋ and define X >h 0 = { x ∈ B ∞ : h ( x ) > h 0 } . W e claim that ther e exists a complete bin ary s u btree T ⊆ B n of heigh t at least ⌈ n/ 3 ⌉ , suc h that we ha v e f ( T ) ⊆ X >h 0 . Indeed, let h min = m in { h ( x ) : x ∈ f ( B n ) } and h max = m ax { h ( x ) : x ∈ f ( B n ) } . If h min > h 0 then f ( B n ) ⊆ X >h 0 , and w e can tak e T = B n . So assume that h min < h 0 . Since f is an injection it must satisfy h max > n . Hence k f k Lip > h max − h min 2 n > n − h 0 2 n > 1 4 . Sin ce dist( f ) 6 s ( n ) 44 w e conclude that k f − 1 k Lip 6 4 s ( n ). It follo ws th at, since diam( X r X >h 0 ) 6 2 h 0 , w e ha ve diam  f − 1 ( X r X >h 0 )  6 8 h 0 s ( n ) 6 n / 5 . If the top ⌈ n/ 3 ⌉ lev els of B n are mapp ed in to X >h 0 then w e are d on e, so assume that there exists u ∈ f − 1 ( X r X >h 0 ) of depth at m ost 6 ⌈ n/ 3 ⌉ . In this case f − 1 ( X r X >h 0 ) must b e con tained in the first ⌈ n/ 3 ⌉ + n/ 5 < 2 n/ 3 − 1 lev els of B n , s o we can tak e T to b e any sub tree of B n con tained in the last ⌈ n/ 3 ⌉ lev els of B n . Fix δ ∈ (0 , 1). By Theorem 1.14 (with t = 4, D = s ( n ) and ξ = δ ), there exists a un iv ersal constan t κ > 0 such that if n > s ( n ) κ/δ then there exists a m ap p ing φ : B 4 → B n with dist( φ ) 6 1 + δ suc h that f ◦ φ is a (1 + δ )-ve rtically faithful embedd ing of B 4 in to X >h 0 . Cho osing δ = κ log s ( n ) log n , b y increasing C in (95) if necessary , we ma y assume th at δ < 1 / 40 0. Lemma 6.27 then imp lies (1 + δ ) dist( f ) > dist( f ◦ φ ) > 1 500 δ + ε h 0 = 1 500 κ log s ( n ) log n + 1 s ( ⌊ n/ (40 s ( n )) ⌋ ) . The deduction of (7) fr om (6) is a simp le exercise: if s ( n ) = o (log n/ log log n ) then w e ha v e ( s ( n ) log s ( n )) / log n = o (1). The desired claim w ill then follo w once w e c heck that lim su p n →∞ s ( ⌊ n/ ( 40 s ( n )) ⌋ ) s ( n ) = 1 . (96) Indeed, if (96) failed then there w ould exist ε 0 ∈ (0 , 1) and n 0 ∈ N suc h that for all n > n 0 , s ( ⌊ n/ log n ⌋ ) 6 s ( ⌊ n/ (4 0 s ( n )) ⌋ ) 6 (1 − ε 0 ) s ( n ) . (97) Iterating (97), it would follo w that s ( n j ) > n Ω(1) j for some subsequ ence { n j } ∞ j =1 , a con tradiction. Pr o of of The or em 1.12. The p ro of is ident ical to the ab o ve argum en t: all o ne has to notice is that when s ( n ) = D for all n ∈ N the resulting metric d ε on B ∞ is D -equiv alen t to the original shortest path metric on B ∞ . In this case, if c X ( B n ) 6 D − ε then the b ound (95) implies that n 6 D C D 2 /ε . 7 Discussion and op en problems A v ery interesting question that arises naturally from Th eorem 1.3 and is also a part of the Rib e program, is fin ding a metric c h aracterizat ion of q -smo othness. A Banac h space ( X , k · k X ) is called q -smo oth if it admits an equiv alen t norm | | | · | | | s u c h that th ere is a constan t S > 0 satisfying: | | | x | | | = 1 ∧ y ∈ X = ⇒ | | | x + y | | | + | | | x − y | | | 2 6 1 + S | | | y | | | q . A Banac h space X is p -con ve x if and only if its dual space X ∗ is q -smo oth, where 1 p + 1 q = 1 [17]. It is kn o w n that a Banac h space X is p -conv ex for some p < ∞ (i.e., s up er r eflexiv e) if and only if it is q -smo oth for some q > 1 (this follo w s from [10, 29]). Hence Bourgain’s metric c haracterizatio n of sup erreflexivit y can b e viewe d as a statemen t ab out uniform smo othn ess as well. H o w ev er , we still lac k a metric charac terization of the more useful notion of q -smo othness. T rees are natur al candidates for fin ite metric obstructions to p -con vexit y , but it is unclear what w ould b e the p ossible finite metric witnesses to the “non- q -smo othness” of a metric space. 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