Central and $L^p$-concentration of 1-Lipschitz maps into $mathbb{R}$-trees

CENTRAL AND L p -CONCENTRA TION OF 1 -LIP SCHITZ MAPS INTO R -TR EES KEI FUNANO Abstract. In this pap er, we study the L´ evy-Milman concentration phenomeno n o f 1-Lipschitz maps fro m mm-spaces to R -tr ees. O ur main theorems assert that the con- centration to R -trees is equiv alent to the concentration to the real line. 1. Intr oduction This pap er is dev oted to in ves tigating the L´ evy-Milman concen t r a tion phenomenon of 1-Lipsc hitz maps fr o m mm-spaces (metric measure spaces) to R -trees. Here, an mm- space is a triple ( X, d X , µ X ) o f a set X , a complete separable distance function d X on X , and a finite Borel measure µ X on ( X , d X ). Let { ( X n , d X n , µ X n ) } ∞ n =1 b e a sequence o f mm-spaces and { ( Y n , d Y n ) } ∞ n =1 a sequence o f metric spaces. G iven a sequence { f n : X n → Y n } ∞ n =1 of 1-Lipsc hitz maps, w e consider the follow ing three prop erties: (i) (Concen tration prop erty ) There exist p oints m f n ∈ Y n , n ∈ N , suc h that µ X n  { x n ∈ X n | d Y n ( f n ( x n ) , m f n ) ≥ ε }  → 0 as n → ∞ for an y ε > 0. (ii) (Cen tral concen tration prop erty) The maps f n , n ∈ N , concen trate to the cen ter of mass o f the push-forw ard measure ( f n ) ∗ ( µ X n ). In other words, t he concen tration prop ert y (i) holds in the case where m f n is the cen ter of mass. (iii) ( L p -concen tratio n prop erty ) F or a n umber p > 0 , we hav e Z Z X n × X n d Y n  f n ( x n ) , f n ( y n )  p dµ X n ( x n ) dµ X n ( y n ) → 0 as n → ∞ Eac h target metric space Y n , n ∈ N , is called a scr e en . Cheb yshev’s inequalit y prov es that the L p -concen tratio n (iii) implies the concen tration prop erty (i) for an y p > 0. If eac h screen Y n , n ∈ N , is an Euclidean space R k , then the L p -concen tratio n (iii) f o r p ≥ 1 yields the central concen tration prop ert y (ii) (see Lemma 2.18) . The cen tral concen tration (ii) is stronger than the concen tra t io n prop erty (i). There is an example of maps f n , n ∈ N , with the concentration prop ert y (i), but not ha ving the cen tral concen tra tion prop ert y Date : August 9, 2021 . 2000 Mathematics Su bje ct Classific ation. 53C21 , 53C23. Key wor ds and phr ases. median, mm-space, obser v able L p -v ariatio n, o bserv able diameter, obs e rv able central radius, R -tree. This work was partially supp o rted by Research F ellowships of the Japa n So ciety for the Pr omotion of Science for Y oung Scien tists. 1 2 KEI FUNANO (ii) ( see Remark 2.17). In some special cases, the concen tration (i) implies the central and L p -concen tratio n prop erties (ii) and (iii) (see [3, Subsection 2.4] and [7, Section 3 1 2 . 31]). Vitali D. Milman first introduced the concen tratio n and the cen tral concen tration prop- erties (i) and (ii) for 1-Lipsc hitz functions (i.e., Y n = R , n ∈ N ) and emphasized their imp ortance in his inv estigatio n of asymptotic geometric analysis (see [11]). No wada ys those prop erties are widely studied b y man y literature a nd blend with v a r ious areas o f mathematics (see [7], [9], [12], [13], [14], [16], [17] and references therein fo r further in- formation). M. Gromo v first considered the case o f general screens in [5], [6], and [7, Chapter 3 1 2 ]. Se e [3], [4], a nd [10] for another w orks o f general screens. In [7], G r o mo v settled the concen tr ation and central concen tration prop erties (i) and (ii) for 1-Lipsc hitz maps by intro ducing the observable d i ameter ObsDiam Y ( X ; − κ ) and the o b servable c en- tr al r adius ObsCRad Y ( X ; − κ ) f or an mm-space X , a metric space Y , a nd κ > 0 (see Sec- tion 2 for the precise definitions). The L 2 -concen tratio n prop erty (iii) w as first app ear ed in Gromo v’s pap er [5]. Motiv ated b y [5], the author in tro duced in [3] the observable L p -variation Obs L p -V ar Y ( X ) to study the prop ert y (iii) (see Section 2 for the defini- tion). Note that given a sequence { X n } ∞ n =1 of mm-spaces and { Y n } ∞ n =1 of metric spaces, ObsDiam Y n ( X n ; − κ ) (resp., ObsCRad Y n ( X n ; − κ ), Obs L p -V ar Y n ( X n )) conv erges to zero as n → ∞ for any κ > 0 if and only if any sequence { f n : X n → Y n } ∞ n =1 of 1- L ipschitz maps (resp., cen tra l, L p -)concen trat es. In this pap er, w e treat the case of R -t r ee screens. Theorem 1.1. L et { X n } ∞ n =1 b e a se quenc e of mm-sp ac es. Then, the fol low ing (1.1) and (1.2) a r e e quiva l e n t to e ach other. ObsDiam R ( X n ; − κ ) → 0 a s n → ∞ for any κ > 0 . (1.1) sup { ObsDiam T ( X n ; − κ ) | T is an R -tr e e } → 0 as n → ∞ fo r a n y κ > 0 . (1.2) Theorem 1.1 is a complete solution to Gromo v’s exercise in [7, Section 3 1 2 . 32]. In [3, Section 5], the author prov ed it only for simplicial tree screens. The implication (1.2) ⇒ (1.1) is ob vious. F or the pro of of the conv erse, w e define the notion of a me d ian for a finite Borel measure on an R -tree in Section 3 a nd pro ve s that an y 1- Lipsc hitz maps f n from X n in to R -trees concen trate to medians for the push-forw ard measure ( f n ) ∗ ( µ X n ). T o study the cen tra l a nd L p -concen tratio n for (ii) and (iii) into R -trees, we estimate the distance b etw een the cen ter of mass and a median of a finite Borel measure on an R -tree from the ab o v e in Section 5. F or this estimate, w e partia lly extend K-T. Sturm’s c haracterization of the cen ter of mass on a simplicial tree to the case of an R -tree (see Prop osition 2 .1 2 and Section 4) . F rom the estimate, w e b ound ObsCRad T ( X ; − κ ) ( r esp., Obs L p -V ar T ( X )) fr o m the ab ov e in terms o f ObsCRad R ( X ; − κ ) (resp., Obs L p -V ar R ( X )) (see Prop ositions 5.5 and 5.7). As a result, w e obtain Theorem 1.2. L et { X n } ∞ n =1 b e a se quenc e of mm-sp ac es. Then, the fol low ing (1.3) and (1.4) a r e e quiva l e n t to e ach other. ObsCRad R ( X n ; − κ ) → 0 a s n → ∞ for any κ > 0 . (1.3) CENTRAL AND L p -CONCENTRA TIO N OF 1-LIPSCHITZ MAPS INTO R -TREES 3 sup { ObsCRad T ( X n ; − κ ) | T is an R -tr e e } → 0 as n → ∞ fo r a n y κ > 0 . (1.4) Theorem 1.3. L e t { X n } ∞ n =1 b e a se quenc e of mm-sp ac es and p ≥ 1 . T hen, the fol lowing (1.5) a n d (1.6) ar e e quivalent to e ach other. Obs L p - V ar R ( X n ) → 0 as n → ∞ . (1.5) sup { Obs L p - V ar T ( X n ) | T is an R -tr e e } → 0 as n → ∞ . (1.6) The condition (1.3) is stronger than (1.1) (see Lemma 2 .1 6 and Remark 2.17), a nd (1.5) implies (1.3) (see Lemma 2 .18). It seems that the conditions (1.3) a nd (1 .5 ) are not equiv alen t, but we hav e no coun terexample. In our previous w o rk, the author inv estigated the ab ov e prop erties (i), (ii), and (iii) fo r 1-Lipsc hitz maps in to Hadamard manifolds (see [3, Theorems 1.3, 1.4, and Lemma 4.4]). The L 2 -concen tratio n prop erty (iii) in that case is a lso studied by Gromov (see [5, Section 13]). Our theorems are though t as of 1-dimensional analogue to these w orks. 2. Preliminaries 2.1. Basics of the concen tr ation and the L p -concen tration. 2.1.1. O bservable diameter and sep ar ation distanc e. Let Y b e a metric space and ν a Borel measure on Y such tha t m := ν ( Y ) < + ∞ . W e define for an y κ > 0 diam( ν, m − κ ) := inf { diam Y 0 | Y 0 ⊆ Y is a Bo rel subset suc h that ν ( Y 0 ) ≥ m − κ } and call it the p artial dia meter of ν . Definition 2.1 (Observ able diameter) . Let ( X , d X , µ X ) b e an mm-space with m := µ X ( X ) and Y a metric space. F or an y κ > 0 w e define the observable diameter of X b y ObsDiam Y ( X ; − κ ) := sup { diam( f ∗ ( µ X ) , m − κ ) | f : X → Y is a 1-Lipsc hitz map } . The target metric space Y is called t he scr e en . The idea of the observ able diameter comes from the quan tum and statistical mec hanics, that is, we think of µ X as a state on a configuration space X a nd f is interpreted a s a n observ able. Let ( X , d X , µ X ) b e an mm-space. F or an y κ 1 , κ 2 ≥ 0, we define the sep ar ation distanc e Sep( X ; κ 1 , κ 2 ) = Sep( µ X ; κ 1 , κ 2 ) of X as the suprem um of the distance d X ( A, B ), where A and B are Bo r el subsets of X satisfying that µ X ( A ) ≥ κ 1 and µ X ( B ) ≥ κ 2 . The pro of of the follo wing lemmas are easy and w e omit t he pro of. Lemma 2.2 (cf. [7, Section 3 1 2 . 33]) . L et ( X , d X , µ X ) and ( Y , d Y , µ Y ) b e two mm-sp ac es. Assume that a 1 -Lipschitz m ap f : X → R satisfies f ∗ ( µ X ) = µ Y . Then we have Sep( Y ; κ 1 , κ 2 ) ≤ Sep( X ; κ 1 , κ 2 ) Lemma 2.3. F or an y κ > m/ 2 , we have Sep( X ; κ, κ ) = 0 . 4 KEI FUNANO The r elationships b etw een the observ able diameter a nd the separation distance are the follo wing: Prop osition 2.4 (cf. [7, Section 3 1 2 . 33]) . L et ( X , d , µ ) b e an mm - s p ac e and 0 < κ ′ < κ . Then w e have Sep( X ; κ, κ ) ≤ ObsDiam R ( X ; − κ ′ ) . Prop osition 2.5 (cf. [7, Section 3 1 2 . 33]) . F or any κ > 0 , we hav e ObsDiam R ( X ; − 2 κ ) ≤ Sep( X ; κ, κ ) . See [4 , Subsection 2 . 2] for details of the pro ofs of the ab ov e prop ositions. Corollary 2.6 (cf. [7, Section 3 1 2 . 33]) . A se quenc e { X n } ∞ n =1 of mm- s p ac es satisfie s that ObsDiam R ( X n ; − κ ) → 0 a s n → ∞ for any κ > 0 if and only if Sep( X n ; κ, κ ) → 0 as n → ∞ for any κ > 0 . 2.1.2. O bservable L p -variation. Let ( X , d X , µ X ) b e an mm-space and ( Y , d Y ) a metric space. Giv en a Bor el measure ν on Y and p ∈ (0 , + ∞ ), we put V p ( ν ) :=  Z Z Y × Y d Y ( x, y ) p dν ( x ) d ν ( y )  1 /p . F or a Borel measurable map f : X → Y , w e also put V p ( f ) := V p  f ∗ ( µ X )  Let { X n } ∞ n =1 b e a sequence of mm-spaces and { Y n } ∞ n =1 a sequence of metric spaces. F or an y p ∈ ( 0 , + ∞ ], w e sa y that a sequence { f n : X n → Y n } ∞ n =1 of Borel measurable maps L p -c onc entr ates if V p ( f n ) → 0 as n → ∞ . Giv en an mm-space X and a metric space Y w e define Obs L p -V ar Y ( X ) := sup { V p ( f ) | f : X → Y is a 1-Lipsc hitz map } , and call it the observable L p -variation of X . Lemma 2.7. F or an y cl o s e d subset A ⊂ X , we have Obs L p - V ar R ( A ) ≤ Obs L p - V ar R ( X ) . Pr o of . Let f : A → R b e an arbitrary 1-Lipsc hitz function. F rom [1, Theorem 3.1 .2], there exists a 1-Lipsc hitz extension of f , say e f : X → R . Hence, we g et V p ( f ) ≤ V p ( e f ) ≤ Obs L p -V ar R ( X ) . This completes the pro of.  See [3, Subsection 2.4] f or the relatio nships b et wee n the observ able diameter a nd the observ able L p -v ariation. CENTRAL AND L p -CONCENTRA TIO N OF 1-LIPSCHITZ MAPS INTO R -TREES 5 2.2. Basics of R -trees. Before reviewing the definition of R -trees, w e recall some stan- dard terminologies in metric geometry . Let ( X , d X ) b e a metric space. A rectifiable curv e η : [0 , 1] → X is called a ge o des i c if its arclength coincides with the distance d X  η (0 ) , η (1)  and it has a constant speed, i.e., parameterized prop ortio nally to the a rc length. W e sa y that ( X , d X ) is a ge o desic sp ac e if any t w o p oints in X a r e joined by a g eo desic b etw een them. Let X b e a geo desic space. A ge o desic triang le in X is the union of the ima g e o f three geo desics joining a triple of p oints in X pairwise. A subset A ⊆ X is called c onve x if every geo desic joining t w o p oin ts in A is con ta ined in A . A complete metric space ( T , d T ) is called an R -tr e e if it has t he follo wing prop erties: (1) F or all z , w ∈ T there exists a unique unit sp eed geo desic φ z ,w from z to w . (2) The image of ev ery simple path in T is t he imag e o f a geo desic. Denote by [ z , w ] T the image of the geo desic φ z ,w . W e also put ( z , w ] T := [ z , w ] T \ { z } and ( z , w ) T := [ z , w ] T \ { z , w } . A complete geo desic space T is an R -tr ee if and only if it is 0- hyperb o lic, that is to sa y , ev ery edge in any geo desic t riangle in T is included in the union o f the other t w o edges. See [2] for another characterizations of R -trees. G iv en z ∈ T , we indicate b y C T ( z ) the set of all connected comp onen ts of T \ { z } . W e also denote b y C ′ T ( z ) the set of all { z } ∪ T ′ for T ′ ∈ C T ( z ). Althoug h t he follow ing lemma is somewhat standard, w e prov e it for the completeness. Lemma 2.8. Each T ′ ∈ C T ( z ) is c o n vex. Pr o of . F rom the prop erty (2 ) of R -trees, it is sufficien t to pro v e tha t T ′ is arcwise con- nected. T aking a p oint z ∈ T ′ , we put A := { w ∈ T ′ | z and w are connected by a path in T ′ } . It is easy to see that the set A is closed in T ′ . Since eve ry metric ball in T is a r cwise connected, the set A is also op en. Since T ′ is connected, w e get T ′ = A . This completes the pro of.  A subset in an R -tree is called a subtr e e if it is a closed conv ex subset. Note that a subtree is itself an R -tree. Prop osition 2.9. Every c onne cte d subset i n an R -tr e e is c onvex. Pr o of . Let T b e an R -tree. Supp o se that there exists a connected subset T ′ ⊆ T which is not con v ex. Then, there are p oin ts z , w ∈ T ′ and e z ∈ ( z , w ) T suc h that e z 6∈ T ′ . Since T ′ = S { T ′ ∩ C | C ∈ C T ( e z ) } and eac h C ∈ C T ( e z ) is op en, from the connectivit y of T ′ , there is C 0 ∈ C T ( e z ) suc h that T ′ ⊆ C 0 . Since C 0 is con v ex b y Lemma 2.8, w e get e z ∈ [ z , w ] T ⊆ C 0 . This is a contadiction since e z 6∈ C 0 . This completes the pro of.  2.3. Cen ter of mass of a measure on a CA T(0)-space and observ able cen tral radius. 6 KEI FUNANO 2.3.1. B asics of the c enter of mass of a me asur e on CA T(0 ) - sp ac es. In this subsection, w e review Sturm’s works a b out measures on a CA T(0)-spaces. Refer [8] and [15] for details. A geo desic metric space X is called a CA T(0)-space if w e ha ve d X  x, γ (1 / 2)  2 ≤ 1 2 d X ( x, y ) 2 + 1 2 d X ( x, z ) 2 − 1 4 d X ( y , z ) 2 for any x, y , z ∈ X and an y minimizing geo desic γ : [0 , 1 ] → X from y to z . F or example, Hadamard manifolds, Hilb ert spaces, and R -tr ees ar e all CA T(0) - spaces. Let ( X , d X ) b e a metric space. W e denote b y B ( X ) the set o f all finite Borel measures ν on X with the separable supp ort. W e indicate by B 1 ( X ) the set of a ll Borel measures ν ∈ B ( X ) such that R X d X ( x, y ) d ν ( y ) < + ∞ for some (hence all) x ∈ X . W e also indicate b y P 1 ( X ) the set o f all probability measures in B 1 ( X ). F or a ny ν ∈ B 1 ( X ) and z ∈ X , w e consider the function h z ,ν : X → R defined b y h z ,ν ( x ) := Z X { d X ( x, y ) 2 − d X ( z , y ) 2 } d ν ( y ) . Note that Z X | d X ( x, y ) 2 − d X ( z , y ) 2 | dν ( y ) ≤ d X ( x, z ) Z X { d X ( x, y ) + d X ( z , y ) } dν ( y ) < + ∞ . A p oint z 0 ∈ X is called the c enter of mass of t he measure ν ∈ B 1 ( X ) if for an y z ∈ X , z 0 is a unique minimizing p oin t of the function h z ,ν . W e denote the p oin t z 0 b y c ( ν ). A metric space X is said to b e c entric if eve ry ν ∈ B 1 ( X ) has the cen ter of mass. Prop osition 2.10 (cf. [15, Prop osition 4 . 3]) . A CA T(0 )-sp ac e is c entric. A simple v ariatio nal argumen t yields the follo wing lemma. Lemma 2.11 ( cf. [15, Propsition 5 . 4]) . L e t H b e a Hilb ert sp ac e . Then for e ach ν ∈ B 1 ( H ) with m = ν ( X ) , we h ave c ( ν ) = 1 m Z H y dν ( y ) . Let ( T , d T ) b e an R -tree and ν ∈ B 1 ( T ). F or z ∈ T and T ′ ∈ C ′ T ( z ), we put c z ,T ′ ( ν ) := Z T ′ d T ( z , w ) dν ( w ) − Z T \ T ′ d T ( z , w ) dν ( w ) . Let us consider a (p ossibly infinite) simplicial tree T s . Here, the length of each edge of T s is not necessarily equal t o 1. W e assume that ev ery v ertex of T s is an isolated p oin t in the v ertex set o f T s . Prop osition 2.12 (cf. [15, Prop osition 5 . 9]) . L et ν ∈ B 1 ( T s ) and z ∈ T s . Then, z = c ( ν ) if and on ly i f c z ,T ′ ( ν ) ≤ 0 for any T ′ ∈ C ′ T s ( z ) . Prop osition 2.13 (cf. [15, Prop osition 6 . 1]) . L e t N b e a C A T (0)-sp ac e and ν ∈ B 1 ( N ) . Assume that the supp ort of ν is c ontaine d in a close d c onve x subset K of N . Th e n , we have c ( ν ) ∈ K . CENTRAL AND L p -CONCENTRA TIO N OF 1-LIPSCHITZ MAPS INTO R -TREES 7 Let X b e a metric space. F or µ, ν ∈ P 1 ( X ), w e define the L 1 -Wasserstein dis tanc e d W 1 ( µ, ν ) b etw een µ and ν as the infim um of R X × X d X ( x, y ) dπ ( x, y ), where π ∈ P 1 ( X × X ) runs o ver all c ouplings of µ and ν , that is, the measures π with the prop ery that π ( A × X ) = µ ( A ) and π ( X × A ) = ν ( A ) for an y Borel subset A ⊆ X . Lemma 2.14 (cf. [18, Theorem 7.12]) . A se quenc e { µ n } ∞ n =1 ⊆ P 1 ( X ) c onver ges to µ ∈ P 1 ( X ) with r esp e c t to the distanc e f unc tion d W 1 if and only if the se quenc e { µ n } ∞ n =1 c onver ges w e akly to the me asur e µ and lim n →∞ Z X d X ( x, y ) dµ n ( y ) = Z X d X ( x, y ) dµ ( y ) for some (and then an y) x ∈ X . Theorem 2.15 (cf. [15, Theorem 6 . 3]) . L et N b e a CA T(0)-sp ac e. Given µ , ν ∈ P 1 ( N ) , we hav e d N ( c ( µ ) , c ( ν )) ≤ d W 1 ( µ, ν ) . 2.3.2. O bservable c entr al r ad ius. Let Y b e a metric space and assume that ν ∈ B 1 ( Y ) has the cen ter of mass. W e denote b y B Y ( y , r ) the closed ball in Y cen tered at y ∈ Y and with raidus r > 0. F or a ny κ > 0 , putting m := ν ( Y ), w e define the c entr al r adius CRad( ν, m − κ ) of ν as t he infimum of ρ > 0 suc h that ν  B Y ( c ( ν ) , ρ )  ≥ m − κ . Let ( X , d X , µ X ) b e an mm-space with µ X ∈ B 1 ( X ) and Y a cen tric metric space. F or an y κ > 0, w e define ObsCRad Y ( X ; − κ ) := sup { CRad( f ∗ ( µ X ) , m − κ ) | f : X → Y is a 1-Lipsc hitz map } , and call it the observable c e n tr al r adius of X . Lemma 2.16 (cf. [7, Section 3 1 2 . 31]) . F o r any κ > 0 , we h a ve diam( ν, m − κ ) ≤ 2 CRad( ν, m − κ ) . In p articular, w e get ObsDiam Y ( X ; − κ ) ≤ 2 ObsCRad Y ( X ; − κ ) . Remark 2.17. F rom the a b ov e lemma, we see that the cen tral concen tra t ion implies the concen tration. The con vers e is not true in general. F or example, consider the mm-spaces X n := { x n , y n } with distance function d X n giv en b y d X n ( x n , y n ) := n and with a Borel probabilit y measure µ X n giv en by µ X n ( { x n } ) := 1 − 1 / n and µ X n ( { y n } ) := 1 /n . Then, 1-Lipsc hitz maps f n : X n → R defined b y f n ( x ) := d X n ( x, x n ) satisfy that ( f n ) ∗ ( µ X n )  B R ( c (( f n ) ∗ ( µ X n )) , 1 / 2)  = 0 for an y n ∈ N , whereas ObsDiam R ( X n ; − κ ) → 0 as n → ∞ . Lemma 2.18. L et ν ∈ B 1 ( R n ) with m := ν ( R n ) . Then, for any p ≥ 1 and κ > 0 , we have CRad( ν, m − κ ) ≤ V p ( ν ) ( mκ ) 1 /p . (2.1) 8 KEI FUNANO In the c ase of p = 2 , we also h a ve the b etter estim a te CRad( ν, m − κ ) ≤ V 2 ( ν ) √ 2 mκ . (2.2) Pr o of . W e shall pro v e that ν  R n \ B R n  c ( ν ) , ρ 0  ≤ κ for ρ 0 := V p ( ν ) / ( mκ ) 1 /p . Supp ose that ν  R n \ B R n  c ( ν ) , ρ 0  > κ . F rom Lemma 2.11, w e get Z R n | c ( ν ) − x | p dν ( x ) ≤ V p ( ν ) p m . Hence, from Cheb yshev’s inequalit y , w e see that V p ( ν ) p m = ρ p 0 κ < Z R n | c ( ν ) − x | p dν ( x ) ≤ V p ( ν ) p m , whic h is a contradiction. Therefore, w e obtain ν  B R n ( c ( ν ) , ρ 0 )  ≥ m − κ and so (2.1). Since Z R n | c ( ν ) − x | 2 dν ( x ) = V 2 ( ν ) 2 2 m , the same a rgumen t yields (2.2). This completes the pro of.  Corollary 2.19. L et X b e an mm-sp ac e with µ X ∈ B 1 ( X ) . Then, for an y p ≥ 1 , we hav e ObsCRad R n ( X ; − κ ) ≤ 1 ( mκ ) 1 /p Obs L p - V ar R n ( X ) . (2.3) In the c ase of p = 2 , we also h a ve the b etter estim a te ObsCRad R n ( X ; − κ ) ≤ 1 √ 2 mκ Obs L 2 - V ar R n ( X ) (2.4) Corollary 2.20. L et X b e an m m -sp ac e. Then, for any p ≥ 1 an d κ > 0 , we have Sep( X ; κ, κ ) ≤ 2 ( mκ ) 1 /p Obs L p - V ar R ( X ) . (2.5) In the c ase of p = 2 , we also h a ve Sep( X ; κ, κ ) ≤ r 2 mκ Obs L 2 - V ar R ( X ) . (2.6) Pr o of . Assume first that there is a 1 - Lipsc hitz function f : X → R such tha t f ∗ ( µ X ) 6∈ B 1 ( R ). F rom H¨ older’s inequalit y , w e ha ve R R | x − y | p d f ∗ ( µ X )( y ) = + ∞ for an y x ∈ X . This implies V p ( f ) = + ∞ and so Obs L p -V ar R ( X ) = + ∞ . W e consider the other case that f ∗ ( µ X ) ∈ B 1 ( R ) for an y 1-L ipschitz function f : X → R . Com bining Prop osition 2.4 with Lemma 2.16 and (2.3), w e hav e Sep( X ; κ, κ ) ≤ 2 ( mκ ′ ) 1 /p Obs L p -V ar R ( X ) for an y κ > κ ′ > 0 . Letting κ ′ → κ , w e ha ve (2.5). Replacing (2.3) with (2.4 ) in the ab ov e arg umen t, w e also obta in (2 .6).  CENTRAL AND L p -CONCENTRA TIO N OF 1-LIPSCHITZ MAPS INTO R -TREES 9 3. Existence of a median on an R -tree Let T b e an R -t ree and ν a finite Borel measure on T with m := ν ( T ) < + ∞ . A median of ν is a p oint z ∈ T suc h tha t there exist t w o subtrees T ′ , T ′′ ⊆ T suc h that T = T ′ ∪ T ′′ , T ′ ∩ T ′′ = { z } , ν ( T ′ ) ≥ m/ 3, and ν ( T ′′ ) ≥ m/ 3. The existence of a median of a finite Borel measure on a simplic ial tree is prov ed in [3, Prop osition 5.2 ]. The purp ose of this section is to prov e the existence of a median of a finite Borel measure on an R - tree, whic h is needed for the pro ofs of our main theorems. Although t he pro of of the existence is similar to the pro of for the case of a simplicial tree, w e prov e it for the completeness: Prop osition 3.1. Every finite Bor el me asur e on a n R -tr e e has a me dian. Pr o of . Let ν b e a finite Bo r el measure on an R -tree with m := ν ( T ). Assume that a p oin t z ∈ T satisfies that ν ( T ′ ) < m/ 3 for any T ′ ∈ C ′ T ( z ), then it is easy to c hec k that z is a median of ν . So, we assume that for an y z ∈ T there exists T ( z ) ∈ C ′ T ( z ) suc h that ν ( T ( z )) ≥ m/ 3. If fo r some z ∈ T , there exists T ′ ∈ C ′ T ( z ) \ { T ( z ) } suc h that ν ( T ′ ) ≥ m/ 3, then this z is a median of ν . Thereb y , w e also assume that ν ( T ′ ) < m/ 3 for an y z ∈ T and T ′ ∈ C ′ T ( z ) \ { T ( z ) } . Fixing a p o int z 0 ∈ T , we assume that there exists z ∈ T ( z 0 ) \ { z 0 } such that z 0 ∈ T ( z ). Put t 0 := inf { t ∈ (0 , d T ( z 0 , z )] | z 0 ∈ T ( φ z 0 ,z ( t )) } . Claim 3.2. φ z 0 ,z ( t 0 ) is a me dian of ν . Pr o of . Assume first that t 0 = 0 . Then, ta king a monotone decreasing sequence { t n } ∞ n =1 ⊆ (0 , d T ( z 0 , z )] such that t n → 0 as n → ∞ and z 0 ∈ T ( φ z 0 ,z ( t n )) for an y n ∈ N , w e shall show that T ∞ n =1 T ( φ z 0 ,z ( t n )) ⊆  T \ T ( z 0 )  ∪ { z 0 } . If it is, we conclude that the p oin t z 0 = φ z 0 ,z (0) is a median of ν a s follow s: F ro m the uniqueness of T ( φ z 0 ,z 1 ( t n )), w e ha ve T ( φ z 0 ,z ( t n +1 )) ⊆ T ( φ z 0 ,z ( t n )) for eac h n ∈ N . Th us, w e get ν  T ∞ n =1 T ( φ z 0 ,z ( t n ))  = lim n →∞ ν ( T ( φ z 0 ,z ( t n ))) ≥ m/ 3. Supp ose that t here exists w ∈ T ( z 0 ) \ { z 0 } ∩ T ∞ n =1 T ( φ z 0 ,z ( t n )). Note that ( z 0 , z ] T ∩ ( z 0 , w ] T 6 = ∅ . Actually , suppose t hat ( z 0 , z ] T ∩ ( z 0 , w ] T = ∅ . Then, it follow s from the prop erty (2) of R - t r ees that [ z , w ] T = [ z 0 , z ] T ∪ [ z 0 , w ] T . Especially , w e hav e z 0 ∈ [ z , w ] T . Since T ( z 0 ) \ { z 0 } is con v ex b y virtue o f Lemma 2.8, [ z , w ] T do es not contain the p oint z 0 . This is a contradiction. Th us, there exists t ∈ (0 , d T ( z 0 , z )] such that φ z 0 ,z ( t ) ∈ ( z 0 , z ] T ∩ ( z 0 , w ] T . W e pic k n 0 ∈ N with t n 0 < t . Since w ∈ T ( z 0 ) \ { z 0 } ∩ T ∞ n =1 T ( φ z 0 ,z ( t n )) ⊆ T ( φ z 0 ,z ( t n 0 )) \ { z 0 } , we get φ z 0 ,z ( t ) ∈ ( z 0 , w ] T ⊆ T ( φ z 0 ,z ( t n 0 )) \ { z 0 } . Thereb y , w e get φ z 0 ,z ( t ) ∈ T ( φ z 0 ,z ( t n 0 )) \ { φ z 0 ,z ( t n 0 ) } . Therefore, since z 0 ∈ T ( φ z 0 ,z ( t n 0 )) \ { φ z 0 ,z ( t n 0 ) } and T ( φ z 0 ,z ( t n 0 )) \ { φ z 0 ,z ( t n 0 ) } is conv ex, we obta in φ z 0 ,z ( t n ) ∈ [ z 0 , φ z 0 ,z ( t )] T ⊆ T ( φ z 0 ,z ( t n )) \ { φ z 0 ,z ( t n ) } . This is a contradiction. Therefore, w e hav e T ∞ n =1 T ( φ z 0 ,z ( t n )) ⊆  T \ T ( z 0 )  ∪ { z 0 } . W e consider the other case that t 0 > 0 . T ak e a monotone increasing seque nce { t n } ∞ n =1 ⊆ (0 , + ∞ ) suc h that t n → t 0 as n → ∞ and z 0 6∈ T  φ z 0 ,z ( t n )  for eac h n ∈ N . Then, 10 KEI FUNANO the same pro of in the case of t 0 = 0 implies that ν  T ∞ n =1 T ( φ z 0 ,z ( t n ))  ≥ m/ 3 and T ∞ n =1 T ( φ z 0 ,z ( t n )) ⊆  T \ T ( φ z 0 ,z ( t 0 ))  ∪ { φ z 0 ,z ( t 0 ) } . Therefore, φ z 0 ,z ( t 0 ) is a median of ν . This completes the pro of of the claim.  W e next a ssume that z 0 6∈ T ( z ) fo r any z ∈ T ( z 0 ). W e denote b y Γ the set of all unit sp eed geo desics γ : [0 , L ( γ )] → T ( z 0 ) suc h that γ (0) = z 0 and γ ([ t, L ( γ )]) ⊆ T ( γ ( t )) for an y t ∈ [0 , L ( γ )]. Because of t he assumption, w e easily see Claim 3.3. F or any z ∈ T ( z 0 ) , we ha v e φ z 0 ,z ∈ Γ . Claim 3.4. F or any γ , γ ′ ∈ Γ w i th L ( γ ) ≤ L ( γ ′ ) , we ha v e [ γ (0) , γ ( L ( γ ))] T ⊆ [ γ ′ (0) , γ ′ ( L ( γ ′ ))] T . Pr o of . Suppo se that t 0 := sup { t ∈ [0 , L ( γ )] | [ γ (0) , γ ( t )] T ⊆ [ γ ′ (0) , γ ′ ( L ( γ ′ ))] T } < L ( γ ) . Then, w e ha v e γ ( t ) 6∈ [ γ ′ (0) , γ ′ ( L ( γ ′ ))] T for any t > t ′ . Actually , if γ ( t ) ∈ [ γ ′ (0) , γ ′ ( L ( γ ′ ))] T , then we hav e γ ( t ) = γ ′ ( t ). Th us, [ γ ( t 0 ) , γ ( t )] T = [ γ ′ ( t 0 ) , γ ′ ( t )] T b y the pro p ert y (2) of the R -trees. Thereb y , w e get [ γ (0) , γ ( t )] T ⊆ [ γ ′ (0) , γ ′ ( L ( γ ′ ))] T . Since t > t 0 , this contradicts the definition of t 0 . Therefore, from the prop erty (2) of R -trees, w e hav e [ γ ( L ( γ )) , γ ′ ( L ( γ ))] T = [ γ ( t 0 ) , γ ( L ( γ ))] T ∪ [ γ ′ ( t 0 ) , γ ′ ( L ( γ ))] T . (3.1) Since γ , γ ′ ∈ Γ, we ha v e γ ( L ( γ )) , γ ′ ( L ( γ )) ∈ T ( γ ( t 0 )) \ { γ ( t 0 ) } . So, fro m the con v exit y of T ( γ ( t 0 )) \ { γ ( t 0 ) } , w e get [ γ ( L ( γ )) , γ ′ ( L ( γ ′ ))] T ⊆ T ( γ ( t 0 )) \ { γ ( t 0 ) } . This is a con t r adition, b ecause γ ( t 0 ) ∈ [ γ ( L ( γ )) , γ ′ ( L ( γ ′ ))] T from (3.1). This completes the pro of o f the claim.  Putting α := sup { L ( γ ) | γ ∈ Γ } , w e shall sho w that α < + ∞ . If α < + ∞ , w e finish the pro of of the prop o sition a s follows: F ro m the completness of R -trees and Claim 3.4, there exists a unique γ ∈ Γ with L ( γ ) = α . W e also note that α > 0 b y Claim 3.3. Th us, there exists a monotone increasing sequence { t n } ∞ n =1 of p o sitiv e num b ers suc h that t n → α a s n → ∞ . W e easily see that T ( γ ( t n +1 )) ⊆ T ( γ ( t n )) for any n ∈ N and T ∞ n =1 T ( γ ( t n )) = { γ ( L ( γ )) } . Since ν  T ( γ ( t n ))  ≥ m/ 3, the p o in t γ ( L ( γ )) is a median of ν . Supp ose that α = + ∞ . Then, taking a sequence { γ n } ∞ n =1 ⊆ Γ suc h that L ( γ n ) < L ( γ n +1 ) for an y n ∈ N a nd L ( γ n ) → + ∞ as n → ∞ , w e obtain T ∞ n =1 T ( γ n ( L ( γ n ))) = ∅ . Since T ( γ n ( L ( γ n ))) ⊆ T ( γ n +1 ( L ( γ n +1 ))) for a n y n ∈ N , w e hav e 0 = ν  ∞ \ n =1 T ( γ n ( L ( γ n )))  = lim n →∞ ν  T ( γ n ( L ( γ n )))  ≥ m 3 , whic h is a contradiction. This completes the pro of of the prop osition.  CENTRAL AND L p -CONCENTRA TIO N OF 1-LIPSCHITZ MAPS INTO R -TREES 11 4. The nece ssity of Proposition 2.12 f or R -tre es In order to prov e the main theorems, we extend the necessit y of Prop osition 2.12 for R -trees: Prop osition 4.1. L et T b e an R -tr e e and ν ∈ B 1 ( T ) . Then, we have c c ( ν ) ,T ′ ( ν ) ≤ 0 for any T ′ ∈ C ′ T ( c ( ν )) . Pr o of . F or simplicities , w e assume that ν ( T ) = 1. W e shall approximate the measure ν by a measure whose supp ort lies on a simplicial tree in T . G iv en n ∈ N , there exists a compact subset K n ⊆ T such that ν ( T \ K n ) < 1 /n and R T \ K n d T ( c ( ν ) , w ) dν ( w ) < 1 / n . T ake a (1 /n )-net { z n i } l n i =1 of K n with mutually different elemen ts suc h tha t d T ( c ( ν ) , z n 1 ) < 1 /n . W e then tak e a sequenc e { A n i } l n i =1 of m utually disjoint Borel subset of K n suc h that z n i ∈ A n i , diam A n i ≤ 1 /n , and K n = S l n i =1 A n i . Define the Borel probability measure ν n on { z n i } l n i =1 b y ν ( { z n 1 } ) := ν ( A n 1 ) + ν ( T \ K n ) and ν ( { z n i } ) := ν ( A n i ) f or i ≥ 2. Claim 4.2. d W 1 ( ν n , ν ) → 0 as n → ∞ . Pr o of . W e shall sho w that lim n →∞ Z T d T ( c ( ν ) , w ) dν n ( w ) = Z T d T ( c ( ν ) , w ) dν ( w ) . (4.1) Since Z T d T ( c ( ν ) , w ) dν n ( w ) = l n X i =1 d T ( c ( ν ) , z n i ) ν ( A n i ) + d T ( c ( ν ) , z n 1 ) ν ( T \ K n ) , w e ha ve    Z T d T ( c ( ν ) , w ) dν n ( w ) − l n X i =1 d T ( c ( ν ) , z n i ) ν ( A n i )    < 1 n . (4.2) F rom diam A n i < 1 /n , w e get    l n X i =1 d T ( c ( ν ) , z n i ) ν ( A n i ) − Z K n d T ( c ( ν ) , w ) dν ( w )    (4.3) =    l n X i =1 Z A n i  d T ( c ( ν ) , w ) − d T ( c ( ν ) , z n i )  dν ( w )    ≤ l n X i =1 Z A n i d T ( w , z n i ) dν ( w ) < 1 n . Hence, com bining (4.2) with (4.3) and    Z K n d T ( c ( ν ) , w ) dν ( w ) − Z T d T ( c ( ν ) , w ) dν ( w )    ≤ Z T \ K n d T ( c ( ν ) , w ) dν ( w ) < 1 n , w e obtain (4.1). The same w a y of the ab o v e pro of sho ws that the sequence { ν n } ∞ n =1 con ve rges w eakly to the measure ν . Therefore, by using Lemma 2.14, this completes the pro of of the claim.  12 KEI FUNANO Applying Claim 4.2 to Theorem 2.15, w e get c ( ν n ) → c ( ν ) as n → ∞ . Since the con ve x hull in T of the set { z n i } l n i =1 is a simplicial tree with finite ve rtex set and c ( ν n ) is con tained in the con v ex hull by Prop osition 2.13, it follows from Pro p osition 2.12 that c e T ,c ( ν n ) ( ν n ) ≤ 0 fo r a n y e T ∈ C ′ T ( c ( ν n )). Let T ′ ∈ C ′ T ( c ( ν )) . Assume first that c ( ν n ) ∈ T \ T ′ for infinitely man y n ∈ N . Then, taking T n ∈ C ′ T ( c ( ν n )) with T ′ ⊆ T n , w e hav e c T ′ ,c ( ν ) ( ν n ) ≤ c T n ,c ( ν n ) ( ν n ) + d T ( c ( ν n ) , c ( ν )) ≤ d T ( c ( ν n ) , c ( ν )) . Therefore, w e obta in c T ′ ,c ( ν ) ( ν ) = lim n →∞ c T ′ ,c ( ν ) ( ν n ) ≤ 0. W e consider the other case that c ( ν n ) ∈ T ′ for an y n ∈ N . Let z n ∈ [ c ( ν ) , c ( ν 1 )] T b e the unique p oin t suc h that d T ( z n , c ( ν n )) = inf { d T ( z , c ( ν n )) | z ∈ [ c ( ν ) , c ( ν 1 )] T } . By ta king a subseq uence, w e may assume that d T ( c ( ν ) , z n +1 ) ≤ d T ( c ( ν ) , z n ) for a ny n ∈ N . F or eac h n ≥ 2, w e tak e T n ∈ C ′ T ( z n ) and e T n ∈ C ′ T ( c ( ν n )) suc h that c ( ν 1 ) ∈ T n and c ( ν 1 ) ∈ e T n . Observ e tha t T n ⊆ T n +1 . Since T n ⊆ e T n , w e ha ve c T n ,z n ( ν n ) ≤ c e T n ,c ( ν n ) ( ν n ) + d T ( c ( ν n ) , z n ) ≤ d T ( c ( ν n ) , z n ) . (4.4) W e also easily see Claim 4.3. T ′ \ { c ( ν ) } = S ∞ n =2 T n . The same pro of of Claim 4.2 implies that sup n    Z A d T ( z n , w ) dν n ( w ) − Z A d T ( z n , w ) dν ( w )    | A ⊆ T is a Borel subset o → 0 as n → ∞ . Com bining this with (4.4) and Claim 4.3, w e obta in c T ′ ,c ( ν ) ( ν ) = lim n →∞ c T n ,z n ( ν ) = lim n →∞ c T n ,z n ( ν n ) ≤ lim n →∞ d T ( c ( ν n ) , z n ) = 0 . This completes the pro of of the prop osition.  The a uthor do es not kno w whether the con verse of Prop osition 4.1 holds or not. 5. Proof of the main theorems Com bining Prop o sition 3.1 with the same pro of of [3, Lemma 5.3] implies the following prop osition: Prop osition 5.1. L et T b e an R -tr e e and ν a finite Bor e l me asur e. Then, for any κ > 0 , we hav e ν  B T  m ν , Sep  ν ; m 3 , κ 2  ≥ m − κ, (5.1) wher e m ν is a me dian o f the me asur e ν . I n p articular, letting X b e an m m-sp ac e, we have ObsDiam T ( X ; − κ ) ≤ 2 Sep  X ; m 3 , κ 2  . (5.2) CENTRAL AND L p -CONCENTRA TIO N OF 1-LIPSCHITZ MAPS INTO R -TREES 13 Prop osition 5 .1 tog ether with Corollary 2.6 yields Theorem 1.1 . The following w ay to pro v e Theorem 1.1 is muc h easier and more straightforw ard than the ab ov e w ay , that is, to pro ve the existence of a median of a measure on R -trees. Pr o of o f The or em 1.1. Our goal is to prov e the follo wing inequalit y: ObsDiam T ( X ; − κ ) ≤ 2 Sep  X ; κ 3 , κ 3  + 4 ObsDiam R ( X ; − κ ) (5.3) for an y κ > 0. Let f : X → T b e an arbitra ry 1-Lipsc hitz map. Fixing a p o in t z 0 ∈ T , w e shall consider the function g : T → R defined b y g ( z ) := d T ( z , z 0 ). Since g ◦ f : X → R is the 1-Lipschitz function, from the definition o f the observ able diameter, there is an interv a l A = [ s, t ] ⊆ [0 , + ∞ ) suc h that dia m A ≤ O bsDiam T ( X ; − κ ) and ( g ◦ f ) ∗ ( µ X )( A ) ≥ m − κ . Observ e that the set g − 1 ( A ) is the annulus { z ∈ T | s ≤ d T ( z , z 0 ) ≤ t } . W e denote b y C the set o f all connected comp onen ts of the set g − 1 ( A ) \ { z 0 } . Claim 5.2. Assume that s > 0 . Then, for any T ′ ∈ C , we have diam T ′ ≤ 2 diam A . Pr o of . Giv en a n y z 1 , z 2 ∈ T ′ , w e shall sho w that φ z 0 ,z 1 ( s ) = φ z 0 ,z 1 ( s ). Supp ose that φ z 0 ,z 1 ( s ) 6 = φ z 0 ,z 1 ( s ). Then, putting s 0 := sup { t ∈ [0 , + ∞ ) | φ z 0 ,z 1 ( t ) = φ z 0 ,z 1 ( t ) } , we ha ve s 0 < s . F ro m the definition of s 0 and the prop erty (2) of R -trees, w e ha ve ( φ z 0 ,z 1 ( s 0 ) , z 1 ] T ∩ ( φ z 0 ,z 2 ( s 0 ) , z 2 ] T = ∅ . Therefore, from the pro p ert y (2) of R -trees, w e g et [ z 1 , z 2 ] T = [ φ z 0 ,z 1 ( s 0 ) , z 1 ] T ∪ [ φ z 0 ,z 1 ( s 0 ) , z 2 ] T . Hence, since T ′ is conv ex b y virtue of Prop o sition 2.9, the p oin ts z 1 and z 2 m ust b e included in differen t comp onen ts in C T ( φ z 0 ,z 1 ( s 0 )). This is a con tradiction, since T ′ = S { C ∩ T ′ | C ∈ C T ( φ z 0 ,z 1 ( s 0 )) } and T ′ is connected. Thus , w e hav e φ z 0 ,z 1 ( s ) = φ z 0 ,z 2 ( s ). Consequen t ly , we obtain d T ( z 1 , z 2 ) ≤ d T ( z 1 , φ z 0 ,z 1 ( s )) + d T ( φ z 0 ,z 2 ( s ) , z 2 ) ≤ 2( t − s ) ≤ 2 ObsDiam R ( X ; − κ ) . This completes the pro of of the claim.  Assume first that s ≤ Sep( X ; κ/ 3 , κ/ 3) / 2 . Since ev ery path connecting t w o comp o nen ts in C m ust cross t he p oin t z 0 , b y Claim 5.2, w e hav e diam( f ∗ ( µ X ) , m − κ ) ≤ diam g − 1 ( A ) ≤ Sep  X ; κ 3 , κ 3  + 4 ObsDiam R ( X ; − κ ) . W e consider the ot her case that s > Sep( X ; κ/ 3 , κ/ 3) / 2. Supp ose that f ∗ ( µ X )( T ′ ) < κ/ 3 for an y T ′ ∈ C . Since f ∗ ( µ X )( g − 1 ( A )) ≥ m − κ ≥ κ , w e hav e C ′ ⊆ C suc h tha t κ 3 ≤ f ∗ ( µ X )  [ C ′  < 2 κ 3 . Hence, b y putting C ′′ := C \ C ′ , w e get Sep  X ; κ 3 , κ 3  < d T  [ C ′ , [ C ′′  ≤ Sep  f ∗ ( µ X ); κ 3 , κ 3  ≤ Sep  X ; κ 3 , κ 3  , whic h is a con tradiction. Thereb y , there exists T ′ ∈ C suc h t ha t f ∗ ( µ X )( T ′ ) ≥ κ/ 3. F or a subset A ⊆ T and r > 0, w e put A r := { z ∈ T | d T ( z , A ) ≤ r } . 14 KEI FUNANO Claim 5.3. f ∗ ( µ X )  ( T ′ ) Sep( X ; κ/ 3 ,κ/ 3)  ≥ m − 2 κ/ 3 . Pr o of . Suppo se t ha t f ∗ ( µ X )  ( T ′ ) Sep( X ; κ/ 3 ,κ/ 3)  < m − 2 κ/ 3. Then, we ha ve a con tra diction since Sep  X ; κ 3 , κ 3  < d T  T ′ , T \ ( T ′ ) Sep( X ; κ / 3 ,κ/ 3)+ ε  ≤ Sep  f ∗ ( µ X ); κ 3 , κ 3  ≤ Sep  X ; κ 3 , κ 3  for an y sufficien tly small ε > 0.  Com bining Claims 5.2 with 5.3, w e obtain diam( f ∗ ( µ X ) , m − κ ) ≤ diam  ( T ′ ) Sep( X ; κ/ 3 ,κ/ 3)  ≤ 2 Sep  X ; κ 3 , κ 3  + 2 ObsDiam R ( X ; − κ ) and so (5 .3). This completes the pro of of the theorem.  Note that the inequalit y (5.3) yields sligh tly worse estimate for the observ able diameter ObsDiam T ( X ; − κ ) than (5.2). Let T b e an R -tr ee and ν ∈ B 1 ( T ) with m := ν ( X ). T aking a median m ν ∈ T o f the measure ν , w e let T ν an elemen t in C ′ T  c ( ν )  with m ν ∈ T ν . W e t hen define the function ϕ ν : T → R by ϕ ν ( w ) := d T ( z , w ) if w ∈ T ν and ϕ ν ( w ) := − d T ( z , w ) otherwise. The function ϕ ν is clearly the 1- Lipsc hitz function. Lemma 5.4. L et T b e an R -tr e e and ν ∈ B 1 ( T ) . Th en, the function ϕ ν : T → R satisfies that c (( ϕ ν ) ∗ ( ν )) ≤ 0 , | c (( ϕ ν ) ∗ ( ν )) | ≤ CRad(( ϕ ν ) ∗ ( ν ) , m − κ ) + Sep  ( ϕ ν ) ∗ ( ν ); m 3 , κ 2  (5.4) + Sep (( ϕ ν ) ∗ ( ν ); m − κ, m − κ ) , and CRad( ν, m − κ ) ≤ CRad (( ϕ ν ) ∗ ( ν ) , m − κ ) + Sep  ν ; m 3 , κ 2  (5.5) + Sep  ( ϕ ν ) ∗ ( ν ); m 3 , κ 2  + Sep (( ϕ ν ) ∗ ( ν ); m − κ, m − κ ) for any κ > 0 . Pr o of . Com bining Lemma 2.1 1 with Prop osition 4.1, w e ha v e ν ( T ) c (( ϕ ν ) ∗ ( ν )) = Z T ϕ ν ( z ) d ν ( z ) = Z T ν d T ( c ( ν ) , z ) d ν ( z ) − Z T \ T ν d T ( c ( ν ) , z ) d ν ( z ) = c T ν ,c ( ν ) ( ν ) ≤ 0 . Put r 1 := CRad (( ϕ ν ) ∗ ( ν ) , m − κ ) and r 2 := Sep(( ϕ ν ) ∗ ( ν ); m/ 3 , κ/ 2). F rom (5.1 ) , w e observ e that ( ϕ ν ) ∗ ( ν )  B R ( ϕ ν ( m ν ) , r 2 )  ≥ ν  B T ( m ν , r 2 )  ≥ m − κ . Th us, we get d R  B R  c (( ϕ ν ) ∗ ( ν )) , r 1  , B R ( ϕ ( m ν ) , r 2 )  ≤ Sep (( ϕ ν ) ∗ ( ν ); m − κ, m − κ ) (5.6) and so (5 .4). The ab ov e inequality (5.6) together with c (( ϕ ν ) ∗ ( ν )) ≤ 0 yields that d T ( c ( ν ) , m ν ) = ϕ ν ( m ν ) ≤ | c (( ϕ ν ) ∗ ( ν )) − ϕ ν ( m ν ) | ≤ r 1 + r 2 + Sep(( ϕ ν ) ∗ ( ν ); m − κ, m − κ ) =: r 3 . CENTRAL AND L p -CONCENTRA TIO N OF 1-LIPSCHITZ MAPS INTO R -TREES 15 Therefore, putting r 4 := Sep ( ν ; m/ 3 , κ/ 2), w e o btain ν  B T ( c ( ν ) , r 3 + r 4 )  ≥ ν  B T ( m ν , r 4 )  ≥ m − κ and so (5 .5). This completes the pro of .  Prop osition 5.5. L et T b e an R -tr e e an d X a n mm-sp a c e with µ X ∈ B 1 ( X ) . The n , for any κ > 0 w e have ObsCRad T ( X ; − κ ) ≤ ObsCRad R ( X ; − κ ) + 2 Sep  X ; m 3 , κ 2  + Sep ( X ; m − κ, m − κ ) . Pr o of . This follo ws from Lemma 2.2 and Lemma 5.4.  Pr o of o f The or em 1.2. Prop osition 5.5 together with Corollary 2.6 and Lemma 2.16 di- rectly implies the pro of of the theorem.  Lemma 5.6. L et T b e an R -tr e e and ν ∈ B 1 ( T ) . Then, for a n y p ≥ 1 and κ > 0 , we have V p ( ν ) ≤ 2 m 2 /p n CRad(( ϕ ν ) ∗ ( ν ) , m − κ ) + Sep  ( ϕ ν ) ∗ ( ν ); m 3 , κ 2  (5.7) + Sep (( ϕ ν ) ∗ ( ν ); m − κ, m − κ ) o + 2 V p ( ϕ ν ) . In the c ase of p = 2 , we also h a ve the b etter etima te V 2 ( ν ) 2 ≤ 4 m 2 n CRad(( ϕ ν ) ∗ ( ν ) , m − κ ) + Sep  ( ϕ ν ) ∗ ( ν ); m 3 , κ 2  (5.8) + Sep (( ϕ ν ) ∗ ( ν ); m − κ, m − κ ) o 2 + 2 V 2 ( ϕ ν ) 2 . Pr o of . F rom the triangle inequalit y , we hav e V p ( ν ) ≤ 2  Z Z T × T d T ( c ( ν ) , z ) p dν ( z ) dν ( w )  1 /p = 2  m Z T d T ( c ( ν ) , z ) p dν ( z )  1 /p . (5.9) Putting c ν := c (( ϕ ν ) ∗ ( ν )), w e also get  Z T d T ( c ( ν ) , z ) p dν ( z )  1 /p =  Z T | ϕ ν ( z ) | p dν ( z )  1 /p (5.10) ≤ m 1 /p | c ν | +  Z R | c ν − r | p d ( ϕ ν ) ∗ ( ν )( r )  1 /p ≤ m 1 /p | c ν | + V p ( ϕ ν ) m 1 /p , where in the last inequalit y w e used Lemma 2.11. Combining (5.9) with (5.10), w e obta in (5.7). 16 KEI FUNANO In the case of p = 2, w e ha ve Z T d T ( c ( ν ) , z ) 2 dν ( z ) = Z R | r | 2 d ( ϕ ν ) ∗ ( ν )( r ) (5.11) = m | c ν | 2 + Z R | r − c ν | 2 d ( ϕ ν ) ∗ ( ν )( r ) = m | c ν | 2 + V 2 ( ϕ ν ) 2 2 m , where in the second and the la st equalities w e used Lemma 2.11. Substituting (5.1 1 ) to (5.9), w e obtain (5.8). This completes the pro of.  Prop osition 5.7. L et T b e an R -tr e e and X an mm-sp ac e. Then, for an y p ≥ 1 , we have Obs L p - V ar T ( X ) ≤ 2  2 1 /p (1 + 2 · 2 1 /p ) + 1 } Obs L p - V ar R ( X ) . (5.12) In the c ase of p = 2 , we also h a ve the b etter estim a te Obs L 2 - V ar T ( X ) 2 ≤ (38 + 16 √ 2) Obs L 2 - V ar R ( X ) 2 . (5.13) Pr o of . Assume first that f ∗ ( µ X ) ∈ B 1 ( T ) for an y 1-Lipsc hitz map f : X → T . Then, Lemma 2.2 together with Lemma 2.3 and (5.7) implies that Obs L p -V ar T ( X ) ≤ 2 m 2 /p n ObsCRad R ( X ; − κ ) + Sep  X ; m 3 , κ 2 o + 2 Obs L p -V ar R ( X ) ≤ 2 m 2 /p n ObsCRad R ( X ; − κ ) + Sep  X ; κ 2 , κ 2 o + 2 Obs L p -V ar R ( X ) for any 0 < κ < m/ 2 . Hence, applying the inequalities (2.3) and (2.5) to this inequalit y , w e get Obs L p -V ar T ( X ) ≤ 2  m 1 /p κ − 1 /p (1 + 2 · 2 1 /p ) + 1  Obs L p -V ar R ( X ) for any 0 < κ < m/ 2. Letting κ → m/ 2, w e g et (5.12) . In the case of p = 2, from (5.8), w e ha ve Obs L 2 -V ar T ( X ) 2 ≤ 4 m 2 n ObsCRad R ( X ; − κ ) + Sep  X ; κ 2 , κ 2 o 2 + 2 Obs L 2 -V ar R ( X ) 2 for any 0 < κ < m/ 2. Therefore, substituting the inequalities (2.4) and (2.6) to t his inequalit y , w e get Obs L 2 -V ar T ( X ) 2 ≤ 2  mκ − 1 (2 √ 2 + 1) 2 + 1  Obs L 2 -V ar R ( X ) 2 for an y 0 < κ < m/ 2. Letting κ → m/ 2, we o btain (5.13). W e consider the ot her case that there exists a 1-Lipsc hitz map f : X → T with f ∗ ( µ X ) 6∈ B 1 ( T ). By using H¨ older’s inequality and F ubini’s theorem, we hav e V p ( f ) = + ∞ . T aking x 0 ∈ X , we put f n := f | B X ( x 0 ,n ) for eac h n ∈ N . F rom Lemma 2.7 a nd the ab ov e pro of, w e ha ve V p ( f n ) ≤ Obs L p -V ar T  B X ( x 0 , n )  ≤ 2 { 2 1 /p (1 + 2 · 2 1 /p ) + 1 } Obs L p -V ar R  B X ( x 0 , n )  ≤ 2 { 2 1 /p (1 + 2 · 2 1 /p ) + 1 } Obs L p -V ar R ( X ) . CENTRAL AND L p -CONCENTRA TIO N OF 1-LIPSCHITZ MAPS INTO R -TREES 17 Since V 2 ( f n ) → V 2 ( f ) = + ∞ a s n → ∞ , this implies Obs L p -V ar R ( X ) = + ∞ . This completes the pro of.  Pr o of o f The or em 1.3. Prop osition 5.7 directly implies the pro of of the theorem..  Ac kno wledgemen ts. The author w ould lik e to express his thanks to Professor T a k ashi Shio ya for his v aluable suggestions and assistances during the preparation o f this pap er. 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