Concentration of 1-Lipschitz maps into an infinite dimensional $ell^p$-ball with $ell^q$-distance function

CONCENTRA TION OF 1 -LIP SCHITZ MAPS INTO AN INFIN I TE DIMENSIONAL ℓ p -BALL WITH THE ℓ q -DIST ANCE FUNCTI ON KEI FUNANO Abstract. In this pap er, we study the L ´ evy-Milman concentration phenomenon of 1- Lipschitz maps in to infinite dimensional metric s paces. Our main theorem asserts that the concentration to an infinite dimensional ℓ p -ball with the ℓ q -distance function for 1 ≤ p < q ≤ + ∞ is equiv alen t to the concentration to the r eal line. 1. Intr oduction This pap er is dev oted to inv estigat ing the L ´ evy-Milman concentration phenomenon of 1-Lipsc hitz maps from mm-space s (metric measure spaces) to infinite dime nsional metric spaces. Here, an mm-s p ac e is a triple ( X , d X , µ X ), where d X is a complete separable metric on a set X and µ X a finite Bor el measure on ( X , d X ). The theory of concen tration of 1- Lipsc hitz functions w as first in tro duced b y V. D. Milman in his in v estigation of asymptotic geometric analysis ([17], [18], [19]). No w ada ys, the theory blend with v arious areas of mathematics, such a s geometry , functional analysis and infinite dimensional in tegration, discrete mathematics and complexity theory , probabilit y theory , and so on (see [16], [21], [22], [24] and the references therein for f urther information). The theory of concen tration of maps in to general metric spaces w as first studied b y M. Gro mo v ([11], [12], [13]). He established the theory by introducing the observ able diameter ObsDia m Y ( X ; − κ ) fo r an mm -space X , a metric space Y , a nd κ > 0 in [13] (see Section 2 for the definition of the o bserv able diameter). Giv en a sequence { X n } ∞ n =1 of mm-spaces and a metric space Y , w e note that lim n →∞ ObsDiam Y ( X n ; − κ ) = 0 for an y κ > 0 if and only if for any sequence { f n : X n → Y } ∞ n =1 of 1- Lipsc hitz maps, there exists a sequence { m f n } ∞ n =1 of p oints in Y suc h that lim n →∞ µ X n ( { x n ∈ X n | d Y ( f n ( x n ) , m f n ) ≥ ε } ) = 0 for any ε > 0. If lim n →∞ ObsDiam R ( X n ; − κ ) = 0 for an y κ > 0, then the sequenc e { X n } ∞ n =1 of mm-spaces is called a L´ evy fam ily . The L´ evy families w ere first in tro duced and analyzed by Gromov and Milman in [10]. In our previous w orks [2], [3], [4], [5], the Date : Nov ember 21, 2 018. 2000 Mathematics Subje ct Classific ation. 5 3C21, 53C23 . Key wor ds and phr ases. mm-space, infinite dimens ional ℓ p -ball, concentration of 1-Lipschitz maps, L´ evy g roup. This work was partially supp orted by Research F ellowships o f the Japan So ciety for the Pro motion of Science for Y o ung Scientists. 1 2 KEI FUNANO author prov ed that if a metric space Y is either a n R -tree, a doubling space, a metric graph, or a Hadamard manifold, then lim n →∞ ObsDiam Y ( X n ; − κ ) = 0 holds for an y κ > 0 and any L´ evy family { X n } ∞ n =1 . T o pro v e these results, w e needed to assume the finiteness of the dimension of the targ et metric spaces. In this pap er, w e treat t he case where the dimension of the target metric space Y is infinite. The a uthor has pro v ed in [1] that if the target space Y is so big that an mm- space X with some homogeneit y prop erty can isometrically b e em b edded in to Y , then its observ able diameter ObsDiam Y ( X ; − κ ) is not close to zero. It seems from this result that the concen tration to an infinite dimensional metric space cannot happ en easily . A main theorem of this pap er is the following. F or 1 ≤ p ≤ + ∞ , w e denote by B ∞ ℓ p an infinite dimensional ℓ p -ball { ( x n ) ∞ n =1 ∈ R ∞ | P ∞ n =1 | x n | p ≤ 1 } and by d ℓ p the ℓ p -distance function. Theorem 1.1. L et { X n } ∞ n =1 b e a se quenc e o f mm-s p ac es and 1 ≤ p < q ≤ + ∞ . Then, the se quenc e { X n } ∞ n =1 is a L´ evy fa mily if and only if lim n →∞ ObsDiam ( B ∞ ℓ p , d ℓ q ) ( X n ; − κ ) = 0 fo r a ny κ > 0 . (1.1) As a result, w e obtain the example of the infinite dimensional targ et metric space suc h that the c oncen tration to the space happens as often as the conce n tration to the real line . The pro o f of the sufficiency of Theorem 1.1 is easy . A. Gournay and M. Tsuk amoto’s observ ations play imp ortan t roles for the pro of of the conv erse ([9], [28]). Answ ering a question of Gro mo v in [14, Section 1.1.4], Tsuk amoto pro v ed in [28] that the “macroscopic” dimension o f the space ( B ∞ ℓ p , d ℓ q ) f or 1 ≤ p < q ≤ + ∞ is finite. Gournay indep enden tly pro v ed it in [9] in the case of q = + ∞ . F o r any p and q with 1 ≤ q ≤ p ≤ + ∞ , we hav e an example of a L ´ evy family whic h do es not satisfy (1.1) (see Prop osition 4.4). As applications of Theorem 1.1, by virtue of [3, Prop ositions 4.3 and 4 .4], w e obtain the f ollo wing corollaries of a L ´ evy group a ction. A L ´ evy group was first in tro duced b y Gromov and Milman in [10]. Let a top ological group G acts on a metric space X . The action is called b ounde d if for an y ε > 0 t here exists a neighborho od U o f the iden tit y elemen t e G ∈ G suc h that d X ( x, g x ) < ε for a n y g ∈ U a nd x ∈ X . Note tha t ev ery b ounded action is con tin uous. W e say that the topo logical gro up G acts on X by uniform isomorphisms if for eac h g ∈ G , the map X ∋ x 7→ g x ∈ X is uniform con tin uous. The action is said to be uniformly e quic ontinuous if for any ε > 0 there exists δ > 0 suc h that d X ( g x, g y ) < ε for ev ery g ∈ G and x, y ∈ X with d X ( x, y ) < δ . Given a subset S ⊆ G and x ∈ X , w e put S x := { g x | g ∈ S } . Corollary 1.2. L et 1 ≤ p < q ≤ + ∞ and a ssume that a L´ evy gr oup G b ounde d ly acts on the metric sp ac e ( B ∞ ℓ p , d ℓ q ) b y uniform isomorphisms. Then for any c omp act subset K ⊆ G and any ε > 0 , ther e exists a p oint x ε,K ∈ B ∞ ℓ p such that diam ( K x ε,K ) ≤ ε . Corollary 1.3. Ther e ar e no n o n-trivial b ounde d unifo rmly e q uic ontinuous a c tion s of a L´ evy g r oup to the metric sp ac e ( B ∞ ℓ p , d ℓ q ) for 1 ≤ p < q ≤ + ∞ . Gromov and Milman p oin t ed out in [10] that the unitary group U ( ℓ 2 ) of t he separa- ble Hilb ert space ℓ 2 with the strong top ology is a L ´ evy gro up. Many concrete examples CONCENTRA TION O F 1-LIPSCHI TZ MAPS I NTO ( B ∞ ℓ p , d ℓ q ) 3 of L ´ evy gro ups are known b y t he w orks o f S. Glasner [8], H. F ursten berg and B. W eiss (unpublished), T. G iordano and V. Pe sto v [6], [7], a nd P esto v [25 ], [26]. F or examples, groups of measurable maps from the standard Leb esgue me asure space to compact groups, unitary g roups of some von Neumann algebras, g roups of measure and me asure-class pre- serving automorphisms of the standard Lebesgue measure space, f ull groups of amenable equiv alence relations, and the isometry gro ups of the univ ersal Urys ohn metric space s are L ´ evy gr oups (see the recen t monograph [2 4] for precise). 2. Preliminaries Let Y b e a metric space and ν a Borel me asure on Y suc h that m := ν ( Y ) < + ∞ . W e define for any κ > 0 diam( ν, m − κ ) := inf { diam Y 0 | Y 0 ⊆ Y is a Borel subset suc h t hat ν ( Y 0 ) ≥ m − κ } and call it the p artial d i a meter of ν . Definition 2.1 (Observ able diameter) . Let ( X , d X , µ X ) b e an mm-space with m X := µ X ( X ) and Y a metric space. F or an y κ > 0 w e define the observable diam eter of X by ObsDiam Y ( X ; − κ ) := sup { diam( f ∗ ( µ X ) , m X − κ ) | f : X → Y is a 1- Lipsc hitz map } , where f ∗ ( µ X ) stands for the push-forw ard measure of µ X b y f . 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 config uration space X and f is in terpreted a s an observ able. Let ( X, d X , µ X ) be an mm-space. F or an y κ 1 , κ 2 ≥ 0, we de fine the sep a r 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 ) := inf { d X ( a, b ) | a ∈ A and b ∈ B } , where A and B are Borel subsets of X satisfying that µ X ( A ) ≥ κ 1 and µ X ( B ) ≥ κ 2 . Lemma 2.2 (cf. [13, Section 3 1 2 . 33]) . L et X and Y b e two mm-sp ac es and α > 0 . Assume that an α -Lipschitz map f : X → Y satisfies f ∗ ( µ X ) = µ Y . The n w e have Sep( Y ; κ 1 , κ 2 ) ≤ α Sep( X ; κ 1 , κ 2 ) . Relationships b etw een the observ able diameter and the separation distance a re follo w- ings. W e refer to [4, Subsection 2.2] for pr ecise pro ofs. Lemma 2.3 (cf. [13, Section 3 1 2 . 33]) . L et X b e an mm-sp ac e and κ, κ ′ > 0 with κ > κ ′ . Then we ha ve ObsDiam R ( X ; − κ ′ ) ≥ Sep ( X ; κ, κ ) . Remark 2.4. In [13, Section 3 1 2 . 33], Lemma 2.3 is stated as κ = κ ′ , but t hat is not true in general. F or example, let X := { x 1 , x 2 } , d X ( x 1 , x 2 ) := 1, and µ X ( { x 1 } ) = µ X ( { x 2 } ) := 1 / 2. Putting κ = κ ′ = 1 / 2, we ha v e ObsDiam R ( X ; − 1 / 2) = 0 and Sep( X ; 1 / 2 , 1 / 2 ) = 1. 4 KEI FUNANO Lemma 2.5 (cf. [13 , Section 3 1 2 . 33]) . L et ν b e a Bor el me asur e on R with m := ν ( R ) < + ∞ . Then, fo r any κ > 0 we have diam( ν, m − 2 κ ) ≤ Sep( ν ; κ, κ ) . In p articular, for any κ > 0 we ha v e ObsDiam R ( X ; − 2 κ ) ≤ Sep ( X ; κ, κ ) . Com bining Lemma 2.3 with Lemma 2 .5, w e obtain the following corollary: Corollary 2.6 (cf. [13, Section 3 1 2 . 33]) . A se quenc e { X n } ∞ n =1 of mm -sp ac es is a L´ evy family if and on l y if lim n →∞ Sep( X n ; κ, κ ) = 0 for any κ > 0 . Lemma 2.7. L et ν b e a finite Bor el me asur e on ( R k , d ℓ p ) with m := ν ( R k ) . T hen for any κ > 0 we have diam( ν, m − κ ) ≤ k 1 /p Sep  ν ; κ 2 k , κ 2 k  . Pr o of. F or i = 1 , 2 , · · · , k , let pr i : R k ∋ ( x i ) k i =1 7→ x i ∈ R b e the pro jection. F or Borel subsets A 1 , A 2 , · · · , A k ⊆ R with (pr i ) ∗ ( ν )( A i ) ≥ κ/k , we ha v e ν ( A 1 × A 2 × · · · × A k ) = ν  k \ i =1 (pr i ) − 1 ( A i )  ≥ m − κ, whic h leads to diam( ν, m − κ ) ≤ diam( A 1 × A 2 × · · · × A k ) ≤ k 1 /p max 1 ≤ i ≤ k diam A i . W e therefore get diam( ν, m − κ ) ≤ k 1 /p max 1 ≤ i ≤ k diam  (pr i ) ∗ ( ν ) , m − κ k  . Com bining this with Lemmas 2.2 and 2.5, we obtain diam( ν, m − κ ) ≤ k 1 /p max 1 ≤ i ≤ k Sep  (pr i ) ∗ ( ν ); κ 2 k , κ 2 k  ≤ k 1 /p Sep  ν ; κ 2 k , κ 2 k  . This completes the pro of.  Lemma 2.8. L et a, b b e two r e al numb ers with a < b . Then, a se quenc e { X n } ∞ n =1 of mm-sp ac es is a L´ evy family if an d only if lim n →∞ ObsDiam [ a,b ] ( X n ; − κ ) = 0 for any κ > 0 . (2.1) Pr o of. The necessit y is obvious . W e shall prov e the con v erse. Supp ose that the sequence { X n } ∞ n =1 with the prop ert y (2.1) is not a L ´ evy family . Then, by Corollar y 2.6, there exists κ > 0 and Borel subsets A n , B n ⊆ X n suc h that µ X n ( A n ) ≥ κ , µ X n ( B n ) ≥ κ , a nd lim sup n →∞ d X n ( A n , B n ) > 0. Define a function f n : X n → R b y f n ( x ) := max { d X n ( x, A n ) + a, b } . Since µ X n ( B n ) ≥ κ and lim sup n →∞ d X n ( A n , B n ) > 0, we ha v e lim sup n →∞ diam(( f n ) ∗ ( µ X n ) , m X n − κ ′ ) > 0 CONCENTRA TION O F 1-LIPSCHI TZ MAPS I NTO ( B ∞ ℓ p , d ℓ q ) 5 for any 0 < κ ′ < κ . Since eac h f n is a 1-Lipsc hitz function, this con tradicts the assumption (2.1). This completes t he pro of .  3. Pr oof of the main the orem T o prov e the main theorem, w e extract from Gournay’s pap er [9] and Tsuk amot o’s pap er [28] their arguments . F or k ∈ N , we identify R k with the subset { ( x 1 , x 2 , · · · , x k , 0 , 0 , · · · ) ∈ R ∞ | x i ∈ R for all i } o f R ∞ . Giv en k ∈ N ∪ {∞} , let S k b e the k -th symmetric g roup. W e consider the group G k := {± 1 } k ⋊ S k . The m ultiplication in G k is giv en by (( ε n ) k n =1 , σ ) · (( ε ′ n ) k n =1 , σ ′ ) := (( ε n ε ′ σ − 1 ( n ) ) k n =1 , σ σ ′ ) . The group G k acts on the space R k b y (( ε n ) k n =1 , σ ) · ( x n ) k n =1 := ( ε n x σ − 1 ( n ) ) k n =1 . Note that this action preserv es the k -dimensional ℓ p -ball B k ℓ p ⊆ B ∞ ℓ p and the ℓ q -distance function d ℓ q . Define a subset Λ k ⊆ B k ℓ p b y Λ k := { x ∈ B k ℓ p | x i − 1 ≥ x i ≥ 0 for all i } . Giv en an arbitrary ε > 0 , we put k ( ε ) := ⌈ (2 /ε ) pq / ( q − p ) ⌉ − 1 , where ⌈ (2 /ε ) pq / ( q − p ) ⌉ denotes the smallest integer which is not less t han (2 /ε ) pq / ( q − p ) . F o r k ≥ k ( ε ) + 1, we define a con tin uous map f k ,ε : Λ k → R k ( ε ) b y f k ,ε ( x ) := ( x 1 − x k ( ε )+1 , x 2 − x k ( ε )+1 , · · · , x k ( ε ) − x k ( ε )+1 , 0 , 0 , · · · ) . F or a n y x ∈ B k ℓ p , taking g ∈ G k suc h that g x ∈ Λ k , w e define F k ,ε ( x ) := g − 1 f k ,ε ( g x ) . This definition of the map F k ,ε : B k ℓ p → B k ℓ p is we ll-defined (see [2 8, Sec tion 2] f or details). Giv en k ∈ N , w e put A k := S g ∈ G ∞ g R k ⊆ R ∞ . Theorem 3.1 (cf. [9, Prop osition 1.3] and [28, Section 2]) . The m ap F k ,ε : B k ℓ p → B k ℓ p satisfies that F k ,ε ( B k ℓ p ) ⊆ A k ( ε ) and d ℓ q ( x, F k ,ε ( x )) ≤ ε 2 (3.1) for any x ∈ B k ℓ p . Lemma 3.2. The m a p F k ,ε : ( B k ℓ p , d ℓ q ) → ( A k ( ε ) , d ℓ q ) is a ( 1 + k ( ε ) 1 /q ) -Lipschitz map. Pr o of. By the definition o f the map F k ,ε , it suffice s t o prov e that the map F := F 2 k ( ε )+2 ,ε : ( B 2 k ( ε )+2 ℓ p , d ℓ q ) → ( B 2 k ( ε )+2 ℓ p , d ℓ q ) is (1 + k ( ε ) 1 /q )-Lipsc hitz. Recall that F ( x ) = ( x 1 − x k ( ε )+1 , x 2 − x k ( ε )+1 , · · · , x k ( ε ) − x k ( ε )+1 , 0 , 0 , · · · , 0 ) for an y x ∈ Λ 2 k ( ε )+2 . W e hence get d ℓ q ( F ( x ) , F ( y )) ≤ d ℓ q ( x, y ) + k ( ε ) 1 /q | x k ( ε )+1 − y k ( ε )+1 | ≤ ( 1 + k ( ε ) 1 /q ) d ℓ q ( x, y ) 6 KEI FUNANO for any x, y ∈ Λ 2 k ( ε )+2 . Since eac h g ∈ G 2 k ( ε )+2 preserv es the distance f unction d ℓ q , the map F is (1 + k ( ε ) 1 /q )-Lipsc hitz on eac h g Λ 2 k ( ε )+2 . Let x, y ∈ B 2 k ( ε )+2 ℓ p b e arbitr ary p oints . Observ e that there exist t 0 := 0 ≤ t 1 ≤ t 2 ≤ · · · ≤ t i − 1 ≤ 1 =: t i and g 1 , g 2 , · · · , g i ∈ G 2 k ( ε )+2 suc h that (1 − t ) x + ty ∈ g j Λ 2 k ( ε )+2 for an y t ∈ [ t j − 1 , t j ]. W e therefore obta in d ℓ q ( F ( x ) , F ( y )) ≤ i X j =1 d ℓ q ( F ((1 − t j − 1 ) x + t j − 1 y ) , F ((1 − t j ) x + t j y )) ≤ (1 + k ( ε ) 1 /q ) i X j =1 d ℓ q ((1 − t j − 1 ) x + t j − 1 y , (1 − t j ) x + t j y ) = (1 + k ( ε ) 1 /q ) d ℓ q ( x, y ) . This completes the pro of.  The follow ing lemma is a key to pro v e Theorem 1.1. Lemma 3.3. L et k ∈ N and { ν n,k } ∞ n =1 b e a se q uenc e of finite Bor el me asur es on ( A k , d ℓ q ) satisfying that lim n →∞ Sep( ν n,k ; κ 1 , κ 2 ) = 0 (3.2) for any κ 1 , κ 2 > 0 . Then, putting m n := ν n,k ( A k ) , we have lim n →∞ diam( ν n,k , m n − κ ) = 0 (3.3) for any κ > 0 . Pr o of. It suffices t o pro v e (3.3) by c ho osing a subsequence. W e shall prov e it by induction for k . F or k = 0, since A 0 = { (0 , 0 , · · · ) } , we ha v e diam( ν n, 0 , m n − κ ) = 0. Assume that (3.3) holds for any sequence { ν n,k − 1 } ∞ n =1 of finite Borel measures o n ( A k − 1 , d ℓ q ) havin g the prop ert y (3.2). Let { ν n,k } ∞ n =1 b e any sequence o f finite Bor el mea- sures on ( A k , d ℓ q ) ha ving the prop ert y (3.2). Since lim n →∞ m n = 0 implies (3.3), w e assume that inf n ∈ N m n > 0. Putting a n := max n Sep  ν n,k ; m n 6 , κ 2  , Sep  ν n,k ; m n 6 , m n 6 o , w e get lim n →∞ a n = 0 by the assumption (3.2) and inf n ∈ N m n > 0. Define subsets B n, 1 and B n, 2 of the set A k b y B n, 1 := ( A k − 1 ) a n ∩ A k and B n, 2 := A k \ B n, 1 , where ( A k − 1 ) a n denotes the closed a n -neigh b orho od of A k − 1 . Since A k = B n, 1 ∪ B n, 2 , either the follow ing (1) or (2) holds: (1) ν n,k ( B n, 1 ) ≥ m n / 2 for any sufficie n tly larg e n ∈ N . (2) ν n,k ( B n, 2 ) ≥ m n / 2 for infinitely man y n ∈ N . W e first cons ider the case (2). W e denote b y C n the set o f all connected comp o nen ts of the set B n, 2 . CONCENTRA TION O F 1-LIPSCHI TZ MAPS I NTO ( B ∞ ℓ p , d ℓ q ) 7 Claim 3.4. Ther e exists C n ∈ C n such that ν n,k ( C n ) ≥ m n / 6 . Pr o of. If ν n,k ( C ) < m n / 6 for all C ∈ C n , then there exists C ′ n ⊆ C n suc h that m n 6 ≤ ν n,k  [ C ′ ∈C ′ n C ′  < m n 3 b ecause of ν n,k ( B n, 2 ) ≥ m n / 2. Putting C ′′ n := C n \ C ′ n , w e therefore obtain 2 a n ≤ d ℓ q  [ C ′ ∈C ′ n C ′ , [ C ′′ ∈C ′′ n C ′′  ≤ Sep  ν n,k ; m n 6 , m n 6  < a n , whic h is a contradiction. This completes the pro o f of the claim.  Claim 3.5. Putting D n := ( C n ) Sep( ν n,k ; m n / 6 ,κ/ 2) ∩ A k , we have ν n,k ( D n ) ≥ m n − κ/ 2 . Pr o of. T ak e any δ > 0. Supp osing that ν n,k (( D n ) δ ) < m n − κ/ 2, b y Claim 3.4 , w e get Sep  ν n,k ; m n 6 , κ 2  < d ℓ q ( C n , A k \ ( D n ) δ ) ≤ Sep  ν n,k ; m n 6 , κ 2  , whic h is a con tradiction. This pro v es that ν n,k (( D n ) δ ) ≥ m n − κ for a n y δ > 0. T ending δ → 0, we obtain the claim.  Observ e that D n is isometrically embbeded in to the ℓ q -space ( R k , d ℓ q ). Com bining Lemma 2.7 and Claim 3.5, w e therefore obtain diam( ν n,k , m n − κ ) ≤ diam( ν n,k | D n , m n − κ ) ≤ diam  ν n,k | D n , ν n,k ( D n ) − κ 2  ≤ k 1 /q Sep  ν n,k | D n ; κ 4 k , κ 4 k  ≤ k 1 /q Sep  ν n,k ; κ 4 k , κ 4 k  → 0 as n → ∞ . This implies ( 3.2). W e next consider the case (1). Putting b n := a n + Sep( ν n,k ; m n / 2 , κ/ 2), as in the pro of of Claim 3.5, w e get ν n,k (( A k − 1 ) b n ∩ A k ) = ν n,k (( B n, 1 ) Sep( ν n,k ; m n / 2 ,κ/ 2) ∩ A k ) ≥ m n − κ 2 . Note that there exists a Borel measurable map f n : ( A k − 1 ) b n ∩ A k → A k − 1 suc h that d ℓ q ( x, f n ( x )) = min { d ℓ q ( x, y ) | y ∈ A k − 1 } ≤ b n (3.4) for an y x ∈ ( A k − 1 ) b n ∩ A k . Put ν n,k − 1 := ( f n ) ∗ ( ν n,k | ( A k − 1 ) b n ∩ A k ). An easy calculation pro v es that Sep( ν n,k − 1 ; κ 1 , κ 2 ) ≤ Sep( ν n,k ; κ 1 , κ 2 ) + 2 b n 8 KEI FUNANO for any κ 1 , κ 2 > 0. By this and the prop erty (3.2) f or ν n,k , the measures ν n,k − 1 on A k − 1 satisfy that lim n →∞ Sep( ν n,k − 1 ; κ 1 , κ 2 ) = 0 for an y κ 1 , κ 2 > 0. By the assumption of the induction, w e therefore g et lim n →∞ diam  ν n,k − 1 , ν n,k − 1 ( A k − 1 ) − κ 2  = 0 for an y κ > 0. By using (3.4), w e finally obtain diam( ν n,k , m n − κ ) ≤ diam( ν n,k − 1 , m n − κ ) + 2 b n ≤ diam( ν n,k − 1 , ν n,k − 1 ( A k − 1 ) − κ/ 2) + 2 b n → 0 as n → ∞ . This completes the pro of o f the lemma.  Pr o of of The or em 1.1. Lemma 2.8 directly implies the sufficiency of Theorem 1.1. W e shall prov e the conv erse. Let { f n : X n → ( B ∞ ℓ p , d ℓ q ) } ∞ n =1 b e any sequence of 1-Lipschitz maps. Giv en an arbit rary ε > 0, w e shall pro v e that diam(( f n ) ∗ ( µ X n ) , m X n − κ ) ≤ 2 ε for any κ > 0 and any sufficien tly larg e n ∈ N . Put k := k ( ε ) and ν n,k := ( F ∞ ,ε ◦ f n ) ∗ ( µ X n ). Since diam(( f n ) ∗ ( µ X n ) , m X n − κ ) ≤ diam ( ν n,k , m X n − κ ) + ε b y (3.1), it suffices to prov e that lim n →∞ diam( ν n,k , m X n − κ ) = 0 . (3.5) Since Lemma 2 .2 together with Corollary 2.6 a nd Lemma 3.2 implies that Sep( ν n,k ; κ 1 , κ 2 ) ≤ (1 + k ( ε ) 1 /q ) Sep( X n ; κ 1 , κ 2 ) → 0 as n → ∞ for an y κ 1 , κ 2 > 0, b y virtue of Lemma 3.3, we obtain (3.5). This completes the pro of.  4. Case of 1 ≤ q ≤ p ≤ + ∞ F or a n mm-space X , we define the c on c en tr ation function α X : (0 , + ∞ ) → R as the suprem um of µ X ( X \ A + r ), where A runs o v er all Borel subsets o f X with µ X ( A ) ≥ m X / 2 and A + r is an op en r -neighborho o d of A . Lemma 4.1 (cf. [3, Corollary 2.6]) . A se quenc e { X n } ∞ n =1 of mm-sp ac e s i s a L ´ evy family if and only if lim n →∞ α X n ( r ) = 0 for a ny r > 0 . Let p ≥ 1 . W e shall consider the ℓ n p -sphere S n ℓ p := { ( x i ) n i =1 ∈ R n | P ∞ i =1 | x i | p = 1 } . W e denote b y µ n,p the cone measure a nd ν n,p the surface measure on S n ℓ p normalized as µ n,p ( S n ℓ p ) = ν n,p ( S n ℓ p ) = 1. In ot her w ords, f or an y Borel subset A ⊆ S n ℓ p , w e put µ n,p ( A ) := 1 L ( B n ℓ p ) · L ( { tx | x ∈ A and 0 ≤ t ≤ 1 } ) , CONCENTRA TION O F 1-LIPSCHI TZ MAPS I NTO ( B ∞ ℓ p , d ℓ q ) 9 where L is the Leb esgue measure on R n . By the w orks of G. Schec htman and J. Zinn [27, Theorems 3.1 and 4.1] a nd R. Lata la and J. O. W o jtaszczyk [15, Theorem 5.31], we obtain α ( S n ℓ p , d ℓ 2 ,µ n,p ) ( r ) ≤ C exp ( − cnr min { 2 ,p } ) . (4.1) This inequalit y for p ≥ 2 is also me n tioned by A. Naor in [23, In tro duction] (see also [15, Prop osition 5 .21]). Lemma 4.2. L et 1 ≤ q ≤ p ≤ + ∞ . Then, w e have α ( S n ℓ p , d ℓ q ,µ n,p ) ( r ) ≤ C exp ( − cn 1+(1 / 2 − 1 /q ) min { 2 ,p } r min { 2 ,p } ) if q < 2 and α ( S n ℓ p , d ℓ q ,µ n,p ) ( r ) ≤ C exp ( − cnr min { 2 ,p } ) if q ≥ 2 . Pr o of. If q < 2 , b y d ℓ q ( x, y ) ≤ n 1 /q − 1 / 2 d ℓ 2 ( x, y ), w e then ha v e α ( S n ℓ p , d ℓ q ,µ n,p ) ( r ) ≤ α ( S n ℓ p , d ℓ 2 ,µ n,p ) ( n 1 / 2 − 1 /q r ) ≤ C exp( − cn 1+(1 / 2 − 1 /q ) min { 2 ,p } r min { 2 ,p } ) . If q ≥ 2, b y d ℓ q ( x, y ) ≤ d ℓ 2 ( x, y ), w e then obtain α ( S n ℓ p , d ℓ q ,µ n,p ) ( r ) ≤ α ( S n ℓ p , d ℓ 2 ,µ n,p ) ( r ) ≤ C exp( − cnr min { 2 ,p } ) . This completes the pro of.  Corollary 4.3. Th e se quenc es { ( S n ℓ p , d ℓ q , µ n,p ) } ∞ n =1 and { ( S n ℓ p , d ℓ q , ν n,p ) } ∞ n =1 ar e b o th L´ evy families for 1 ≤ q ≤ p ≤ + ∞ . Pr o of. Since 1+(1 / 2 − 1 /q ) min { 2 , p } > 0, by Lemmas 4 .1 a nd 4.2, the sequen ce { ( S n ℓ p , d ℓ q , µ n,p ) } ∞ n =1 is a L ´ evy family . By virtue o f [23, Theorem 6], the sequence { S n ℓ p , d ℓ q , ν n,p ) } ∞ n =1 is also a L ´ evy fa mily . This completes the pro of.  Prop osition 4.4. L et 1 ≤ q ≤ p ≤ + ∞ . Then, for any κ wi th 0 < κ < 1 / 2 , we have ObsDiam ( B ∞ ℓ p , d ℓ q ) (( S n ℓ p , d ℓ q , µ ); − κ ) ≥ 2 , wher e µ = µ n,p or µ = ν n,p . Pr o of. L et A ⊆ S n ℓ p b e a Borel subset suc h that µ ( A ) ≥ 1 − κ . Since µ ( A ) = µ ( − A ) > 1 / 2, w e hav e µ ( A ∩ ( − A )) > 0. Hence, there exists x ∈ A suc h that − x ∈ A . Since diam A ≥ d ℓ q ( x, − x ) ≥ d ℓ p ( x, − x ) = 2, we obtain diam( µ, 1 − κ ) = 2. Since the inclusion map from the space ( S n ℓ p , d ℓ q ) t o the space ( B ∞ ℓ p , d ℓ q ) is 1-Lipsc hitz, w e obtain t he conclusion. This completes the pro of.  Com bining Corollar y 4.3 with Prop osition 4.4, we obtain an example of a L´ evy family whic h do es no t satisfy (1.1 ) in the case o f 1 ≤ q ≤ p ≤ + ∞ . Ac kno wledgemen ts. The autho r w ould like to express his thanks t o Professor T ak ashi Shio y a for his v aluable suggestions and assistances during the preparation of this pap er. 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