OBSERVABLE CONCENTRATION OF MM-SPACES INTO HADAMARD MANIFOLDS KEI FUNANO

Abstract. In this paper we study the L´evy-Milman concentration phenomenon of 1Lipschitz maps from mm-spaces to Hadamard manifolds. Our main theorem asserts that the concentration of 1-Lipschitz maps to Hadamard manifolds is equivalent to the concentration of 1-Lipschitz maps to Eucliean spaces.

1. Introduction This paper is devoted to study the L´evy-Milman concentration phonomenon of 1Lipschitz maps from mm-spaces to Hadamard manifolds. Here, an mm-space is a triple (X, dX , µX ), where dX is a complete separable metric on a set X and µX a finite Borel measure on (X, dX ). Let µSn be the volume measure on the n-dimensional unit sphere Sn in Rn+1 normalized as µSn (Sn ) = 1. In 1919 P. L´evy showed the following inequality (1.1) by proving the isoperimetric inequality for the sphere Sn . For any 1-Lipschitz function f : Sn → R and any ε > 0, we have (1.1)

2 /2

µSn ({x ∈ Sn | |f (x) − mf | ≥ ε}) ≤ µSn (Sn \ (Sn−1 )ε ) ≤ 2e−(n−1)ε

,

where mf is some constant determined by the function f and (Sn−1 )ε is the closed εneighborhood of the equater Sn−1 ⊆ Sn . For any fixed ε > 0 the right-hand side of the above inequality converges to zero as n → ∞. This high dimensional concentration phenomenon of 1-Lipschitz functions was extensively used and emphasized by V. Milman in his investigation of asymptotic geometric analysis. He used L´evy’s inequality (1.1) for a short proof of Dvoretzky’s theorem on Euclidean section of convex bodies in [14]. Nowadays, the theory of concentration of 1-Lipschitz functions is widely studied in many literature and blend with various areas of mathematics (see [12], [15], and [16]). In the series of works [7, 8, 9] M. Gromov started the theory of concentration of maps to general metric spaces. He captured the phonomenon of concentration of 1-Lipschitz maps visually by introducing the observable diameter for mm-spaces in [9]. He settled the following definition. Let Y be a complete metric space and ν a finite Borel measure Date: July 24, 2010. 2000 Mathematics Subject Classification. 31C15, 53C21, 53C23. Key words and phrases. concentration of 1-Lipschitz maps, Hadamard manifold, L´evy group. This work was partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. 1

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on Y having separable support with the total measure m. For any κ > 0, we define the partial diameter diam(ν, m − κ) of ν as the infimum of the diameter of Y0 , where Y0 runs over all Borel subsets of Y such that ν(Y0 ) ≥ m − κ. Definition 1.1 (Observable diameter). Let X be an mm-space and Y a metric space. For any κ > 0 we define the observable diameter of X by ObsDiamY (X; −κ) := sup{diam(f∗ (µX ), mX − κ) | f : X → Y is a 1-Lipschitz map}, where f∗ (µX ) stands for the push-forward measure of µX by f and mX is the total measure of X. The target metric space Y is called the screen. The idea of the observable diameter comes from the quantum and statistical mechanics, that is, we think of µ as a state on a configuration space X and f is interpreted as ∞ an observable. Given sequences {Xn }∞ n=1 of mm-spaces and {Yn }n=1 of metric spaces limn→∞ ObsDiamYn (Xn ; −κ) = 0 for any κ > 0 if and only if for any sequence {fn : Xn → ∞ Yn } ∞ n=1 of 1-Lipschitz maps there exists a sequence {mfn ∈ Yn }n=1 of points such that (1.2)

lim µXn ({xn ∈ Xn | dYn (fn (xn ), mfn ) ≥ ε}) = 0

n→∞

for any ε > 0. Following Gromov and Milman [10] a sequence {Xn }∞ n=1 of mm-spaces L´evy family if limn→∞ ObsDiamR (Xn ; −κ) = 0 for any κ > 0 (i.e., Yn = R). L´evy families were analyzed by Gromov and Milman in [10] with applications in the topology and the topological fixed point theorem. The inequality (1.1) implies that the sequence {Sn }∞ evy family. Gromov proved in [9] that if Y is either n=1 of the unit spheres is a L´ a compact metric space or a Euclidean space, then limn→∞ ObsDiamY (Xn ; −κ) = 0 for any κ > 0 and for any L´evy family {Xn }∞ n=1 . He also discussed the case where the screens are Euclidean spaces whose dimensions go to infinity. In [8] he considers and analyzes the questions of isoperimetry of waists and concentration for maps from a unit sphere to a Euclidean space. In [13] M. Ledoux and K. Oleszkiewicz estimated the observable diameter ObsDiamRk (X; −κ) provided that the mm-space X has a Gaussian concentration. The author proved that if Y is either a doubling space or an R-tree, then limn→∞ ObsDiamY (Xn ; −κ) = 0 for any L´evy family {Xn }∞ n=1 in [1, 3]. In this paper, inspired by Gromov’s study, we study the concentration phenomenon of 1-Lipschitz maps into Hadamard manifolds, i.e., complete simply connected Riemannian manifolds with nonpositive sectional curvature. We denote by HMn the set of all ndimensional Hadamard manifolds. A main theorem of this paper is the following. ∞ Theorem 1.2. Let {Xn }∞ n=1 be a sequence of mm-spaces and {a(n)}n=1 a sequence of natural numbers. Then, the follwoing (1.3) and (1.4) are equivalent to each other.

(1.3) (1.4)

lim ObsDiamRa(n) (Xn ; −κ) = 0 for any κ > 0.

n→∞

lim sup{ObsDiamN (Xn ; −κ) | N ∈ HMa(n) } = 0 for any κ > 0.

n→∞

OBSERVABLE CONCENTRATION OF MM-SPACES INTO HADAMARD MANIFOLDS

3

The implication (1.4) ⇒ (1.3) is obvious. For the proof of the converse, we find a point in a screen N which is a kind of the center of mass of the push-forward measure on N , and prove that the measure concentrates to the point by the delicate discussions comparing N with both an Euclidean space and a real hyperbolic space. Since the concentration of (1-Lipschitz) functions implies the concentration of maps to Rk for a fixed k ∈ N, we conclude that the concentration of functions implies the concentration of maps to an Hadamard manifold. Recently T. Kondo constructed a twodimensional metric space N of non-positive sectional curvature (so called a CAT(0)-space) and a L´evy family {Xn }∞ n=1 such that lim inf n→∞ ObsDiamN (Xn ; −κ0 ) > 0 for some κ0 > 0 ([11]). Therefore we cannot replace Hadamard manifolds to metric spaces of non-positive sectional curvature in general. The proof of Theorem 1.2 applies only for the smooth case. Observe that the concentration to the real line leads to the concentration to Euclidean spaces Ra(n) for some natural numbers a(n) going to infinity as n grows. In this case, from our theorem, we obtain the concentration to Hadamard manifolds having the same dimensions a(n) as well. As applications of Theorem 1.2, by virtue of [2, Propositions 4.3 and 4.4], we obtain the following corollaries with respect to actions of L´evy groups on Hadamard manifolds. L´evy groups are topologial groups introduced by Gromov and Milman in [10]. Let a topological group G act on a metric space X. The action is called bounded if for any ε > 0 there exists a neighborhood U of the identity element eG ∈ G such that dX (x, gx) < ε for any g ∈ U and x ∈ X. Note that every bounded action is continuous. We say that the topological group G acts on X by uniform isomorphisms if for each g ∈ G, the map X 3 x 7→ gx ∈ X is uniform continuous. The action is said to be uniformly equicontinuous if for any ε > 0 there exists δ > 0 such that dX (gx, gy) < ε for every g ∈ G and x, y ∈ X with dX (x, y) < δ. Given a subset S ⊆ G and x ∈ X, we put Sx := {gx | g ∈ S}. Corollary 1.3. Assume that a L´evy group G boundedly acts on an Hadamard manifold N by uniform isomorphisms. Then for any compact subset K ⊆ G and any ε > 0, there exists a point xε,K ∈ N such that diam(Kxε,K ) ≤ ε. Corollary 1.4. There are no non-trivial bounded uniformly equicontinuous actions of a L´evy group to an Hadamard manifold. Gromov and Milman pointed out in [10] that the unitary group U (`2 ) of the separable Hilbert space `2 with the strong topology is a L´evy group. Many other concrete examples of L´evy groups are known by the works of S. Glasner [6], H. Furstenberg and B. Weiss (unpublished), T. Giordano and V. Pestov [4, 5], and Pestov [18, 19] (see the recent monograph [17] for details). 2. Preliminaries For a measure space (X, µ) with µ(X) < +∞, we denote by F(X, Rk ) the space of all maps from X to Rk . Given λ ≥ 0 and f, g ∈ F(X, Rk ), we put
meλ (f, g) := inf{ε > 0 | µ {x ∈ X | |f (x) − g(x)| ≥ ε} ≤ λε}.

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Note that this meλ is a distance function on F(X, Rk ) for any λ ≥ 0 and its topology on F(X, Rk ) coincides with the topology of the convergence in measure for any λ > 0. The distance functions meλ for all λ > 0 are also mutually equivalent. For an mm-space X, we fix a point x0 ∈ Supp µX . We denote by BX (x0 , r) the closed ball in X centered at x0 with radius r > 0. Lemma 2.1. Let {fn : BX (x0 , n) → Rk }∞ n=1 a sequence of 1-Lipschitz maps. Then, there ∞ k exist a 1-Lipschitz map f : X → R and subsequence {fa(n) }∞ n=1 ⊆ {fn }n=1 such that lim me1 (fa(n) − fa(n) (x0 ), f |BX (x0 ,a(n)) ) = 0.

n→∞

Proof. Since |fn (x) − fn (x0 )| ≤ r for any r > 0 and x ∈ BX (x0 , r), a simple diagonal argument implies the proof of the lemma.  Proposition 2.2. For any κ0 > κ > 0, we have lim inf ObsDiamRk (BX (x0 , n); −κ0 ) ≤ ObsDiamRk (X; −κ). n→∞

Proof. Suppose that lim inf ObsDiamRk (BX (x0 , n); −κ0 ) > α > ObsDiamRk (X; −κ). n→∞

Then, there exists a sequence {fn : BX (x0 , n) → Rk }∞ n=1 of 1-Lipschitz maps such that 0 diam((fn )∗ (µX |BX (x0 ,n) ), mn − κ ) > α for any sufficiently large n ∈ N, where mn := ∞ µX (BX (x0 , n)). According to Lemma 2.1, there exist a subsequence {fa(n) }∞ n=1 ⊆ {fn }n=1 and a 1-Lipschitz map f : X → Rk such that (2.1)

lim me1 (fa(n) − fa(n) (x0 ), f |BX (x0 ,a(n)) ) = 0.

n→∞

Take ε > 0 with 2ε < α − ObsDiamRk (X; −κ) and put An := {x ∈ BX (x0 , a(n)) | |fa(n) (x) − fa(n) (x0 ) − f (x)| < ε}. Assume that a closed subset A ⊆ Rk satisfies that f∗ (µX )(A) ≥ m − κ. Denoting by Bε the closed ε-neighborhood of a subset B ⊆ X, from (2.1), we get (fa(n) )∗ (µX |BX (x0 ,a(n)) )(Aε + fa(n) (x0 )) ≥ f∗ (µX )(A) − µX (X \ An ) ≥ ma(n) − κ0 for any sufficiently large n ∈ N. We therefore obtain diam((fa(n) )∗ (µX |BX (x0 ,a(n)) ), ma(n) − κ0 ) ≤ diam A + 2ε for any sufficiently large n ∈ N, which implies that α < diam((fa(n) )∗ (µX |BX (x0 ,a(n)) ), ma(n) − κ0 ) ≤ diam(f∗ (µX ), m − κ) + 2ε < α for any sufficiently large n ∈ N. This is a contradiction. This completes the proof.



OBSERVABLE CONCENTRATION OF MM-SPACES INTO HADAMARD MANIFOLDS

5

3. Proof of the main theorem Let M be a complete Riemannian manifold and ν a finite Borel measure on M with its finite support. We shall consider the function dν : M → R defined by Z dM (x, y)dν(y), dν (x) := M

where dM is the Riemannian distance on M . The proof of the following lemma is easy and we omit the proof. Lemma 3.1. There exists a point xν ∈ M such that the function dν attains its minimum at the point xν and dM (xν , Supp ν) ≤ diam(Supp ν). Remark 3.2. The above xν is not unique in general. For example, consider M = R and a Borel probability measure ν on R given by ν({0}) = ν({1}) = 1/2. In this case, the function dν attains its infimum at both 0 and 1. From now on, we consider the n-dimensional hyperbolic space Hn as a Poincar´e disk model Dn := {x ∈ Rn | |x| < 1}. For κ1 < 0, we denote by Hn (κ1 ) a complete simply connected Riemannian √ manifold of constant sectional curvature κ1 . We consider n n (H (κ1 ), dHn (κ1 ) ) as (D , (1/ −κ1 ) dHn ). Lemma 3.3 ([20, Theorem 4.6.1]). For any x, y ∈ Hn we have s   |x − y|2 |x − y| n p p + + 1 . (x, y) = 2 log dH (1 − |x|2 )(1 − |y|2 ) 1 − |x|2 1 − |y|2 For each n ∈ N, we define the function φn : Dn → Rn by x φn (x) := . 1 − |x| We shall consider the distance function (φn )∗ dRn on Dn defined by (φn )∗ dRn (x, y) := |φn (x) − φn (y)|. Lemma 3.4. For any x, y ∈ Dn , we have (φn )∗ dRn (x, y) ≥ |x − y|. Proof. Observe that |e x − re y | ≥ |e x − ye| for any r ≥ 1 and any x e, ye ∈ Rn such that |e y | ≥ |e x|. Assuming |x| ≤ |y|, we hence obtain x 1 1 − |x| |x − y| y − y ≥ ≥ |x − y|. (φn )∗ dRn (x, y) = = x − 1 − |x| 1 − |y| 1 − |x| 1 − |y| 1 − |x| This completes the proof. The following theorem is a key to prove Theorem 1.2.



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∞ Theorem 3.5. Let {Xn }∞ n=1 be a sequence of mm-spaces with finite diameters and {a(n)}n=1 ∞ a sequence of natural numbers. We assume that sequences {pn }n=1 of positive numbers satisfy that supn∈N pn < +∞. Then, the following (3.1) and (3.2) are equivalent to each other.   κ lim ObsDiamRa(n) Xn ; − (3.1) = 0 for any κ > 0. n→∞ (diam Xn )pn  o n  κ a(n) (3.2) N ∈ HM = 0 for any κ > 0. lim sup ObsDiamN Xn ; − n→∞ (diam Xn )pn

Combining Theorem 3.5 with Proposition 2.2 implies Theorem 1.2 as follows. Proof of Theorem 1.2. We only prove the implication (1.3) ⇒ (1.4). Let {Nn }∞ n=1 be any a(n) ∞ sequence with Nn ∈ HM and {fn : Xn → Nn }n=1 any sequence of 1-Lipschitz maps. We shall prove that (3.3)

lim diam((fn )∗ (µXn ), mXn − κ) = 0

n→∞

for any κ > 0. By virtue of Proposition 2.2, there is a closed subset An ⊆ Xn such that diam An < +∞, mAn := µXn (An ) ≥ mXn − κ/2, and lim ObsDiamRa(n) (An ; −κ0 ) = 0

(3.4)

n→∞

for any κ0 > 0. The claim (3.3) obviously holds in the case of limn→∞ diam An = 0, so we assume that inf n∈N diam An > 0. Observe that 0 < inf (diam An )1/ diam An ≤ sup(diam An )1/ diam An < +∞. n∈N

n∈N

From this and (3.4), we have  lim ObsDiamRa(n) An ; −

n→∞

 κ0 =0 (diam An )1/ diam An

for any κ0 > 0. Therefore, applying Theorem 3.5 to the sequence {fn |An : An → Nn }∞ n=1 , we obtain diam((fn )∗ (µXn ), mXn − κ) ≤ diam((fn |An )∗ (µXn |An ), mAn − κ/2) → 0 as n → ∞. This completes the proof.



Proof of Theorem 3.5. The implication (3.2) ⇒ (3.1) is obvious. We shall prove the converse. The converse obviously holds in the case of limn→∞ diam Xn = 0, so we assume that inf n∈N diam Xn > 0. Since limn→∞ mXn = 0 implies (3.2), we also assume that a(n) for each n ∈ N inf n∈N mXn > 0. Let {Nn }∞ n=1 be any sequence such that Nn ∈ HM ∞ and {fn : Xn → Nn }n=1 any sequence of 1-Lipschitz maps. Given arbitrary points zn ∈ Nn with d(fn )∗ (µXn ) (zn ) = minz∈Nn d(fn )∗ (µXn ) (z), we shall prove that
κ (fn )∗ (µXn ) Nn \ BNn (zn , r) ≤ (diam Xn )pn

OBSERVABLE CONCENTRATION OF MM-SPACES INTO HADAMARD MANIFOLDS

R a(n)

R a(n)

'en

0

0

exp0

expzn1

D a(n)

zn

7

Nn

0

a(1n)

Figure 1. The map ϕn for any r, κ > 0 and any sufficiently large n ∈ N, which completes the proof of the theorem. Take rn ≥ 1 with fn (Xn ) ⊆ BNn (zn , rn ) and let κn be a negative number such that κn → −∞ as n → ∞ and the sectional curvature on BNn (zn , rn ) is bounded from below by κn . Let ϕ en be a linear isometry from the tangent space of Nn at zn to the a(n) a(n) tangent space of R at 0 and put ϕn := φ−1 en ◦ exp−1 (see z n : Nn → D a(n) ◦ exp0 ◦ϕ Figure 1 below). By the hinge theorem ([21, Chapter IV, Remark 2.6]) and Lemma 3.4, we have |ϕn (x) − ϕn (x0 )| ≤ (φa(n) )∗ dRa(n) (ϕn (x), ϕn (x0 )) ≤ dNn (x, x0 ) for any x, x0 ∈ Nn . Since ϕn ◦ fn is the 1-Lipschitz map from Xn to the Euclidean space (Da(n) , dRa(n) ), from the assumption (3.1) there exists a Borel
subset An ⊆ Da(n) and κ en > 0 such that An ⊆ Supp(ϕn ◦ fn )∗ (µXn ) ⊆ ϕn BNn (zn , rn ) , (ϕn ◦ fn )∗ (µXn )(An ) > mXn − κ en /(diam Xn )pn , limn→∞ κ en = 0, and limn→∞ diam(An , dRa(n) ) = 0. We shall prove that limn→∞ (φa(n) )∗ dRa(n) (An , 0) = 0. If limn→∞ (φa(n) )∗ dRa(n) (An , 0) = 0, then we get An ⊆ ϕn (BNn (zn , r)) for any r > 0 and any sufficiently large n ∈ N. Therefore, we obtain the inequality κ (fn )∗ (µXn )(Nn \ BNn (zn , r)) ≤ (fn )∗ (µXn )(Nn \ ϕ−1 n (An )) ≤ (diam Xn )pn for any r, κ > 0 and any sufficiently large n ∈ N. Suppose that there exists a constant C > 0 such that (φa(n) )∗ dRa(n) (An , 0) ≥ C for any n ∈ N. For any x ∈ Da(n) , we take y ∈ Da(n) such that y = λx as a vector in Ra(n) for

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some λ ≥ 0 and √

1 ∗ dHa(n) (0, y) = (φa(n) ) dRa(n) (0, x). −κn

We define ψn : Da(n) → Da(n) by ψn (x) := y. Since limn→∞ diam(An , dRa(n) ) = 0, limn→∞ κn = −∞, and (φa(n) )∗ dRa(n) (An , 0) ≥ C, we get lim diam(ψn (An ), dRa(n) ) = 0 and lim dRa(n) (ψn (An ), Sa(n)−1 ) = 0.

n→∞

n→∞

√ From this, there exists a point qn ∈ (ψn ◦ ϕn )(BNn (zn , rn )) = BHa(n) (0, −κn rn ) having the following properties (1) and (2). (1) |qn | → 1 as n → ∞. (2) |qn − x| ≤ 2(1 − |qn |) and |qn | ≤ |x| for any x ∈ ψn (An ). Claim 3.6. For any n ∈ N and x ∈ ϕ−1 n (An ) ∩ fn (Xn ), we have −1 dNn (zn , x) ≥ dNn ((ψn ◦ ϕn ) (qn ), x) + bn ,

where bn are some positive numbers satisfying limn→∞ (bn / dNn (zn , (ψn ◦ ϕn )−1 (qn ))) = 1. Proof. According to the hinge √theorem ([21, Chapter IV, Theorem 4.2 (2)]), we have ((ψn ◦ ϕn )−1 (qn ), x) ≤ (1/ −κn ) dHa(n) (qn , (ψn ◦ ϕn )(x)). Note that dNn (zn , x) = dNn√ (1/ −κn ) dHa(n) (0, (ψn ◦ ϕn )(x)). By Lemma 3.3 and (2), we thus have √

−κn dNn (zn , x) −

√

−κn dNn ((ψn ◦ ϕn )−1 (qn ), x)

≥ dHa(n) (0, (ψn ◦ ϕn )(x)) − dHa(n) (qn , (ψn ◦ ϕn )(x)) p   1 − |qn |2 (1 + |(ψn ◦ ϕn )(x)|) p = 2 log |qn − (ψn ◦ ϕn )(x)| + |qn − (ψn ◦ ϕn )(x)|2 + (1 − |(ψn ◦ ϕn )(x)|2 )(1 − |qn |2 ) p  1 − |qn |2 (1 + |(ψn ◦ ϕn )(x)|) 1 p ≥ 2 log 2 1 − |qn | + (1 − |qn |)2 + (1 − |(ψn ◦ ϕn )(x)|2 )(1 − |qn |2 )  1 (1 + |qn |)(1 + |(ψn ◦ ϕn )(x)|) p p = 2 log 2 1 − |qn |2 + 1 − |qn |2 + (1 − |(ψn ◦ ϕn )(x)|2 )(1 + |qn |)2  √ 1 (1 + |qn |)2 p p ≥ 2 log
=: −κn bn . 2 2 2 1 − |qn | 1 + 1 + (1 + |qn |) This completes the proof of the claim.



OBSERVABLE CONCENTRATION OF MM-SPACES INTO HADAMARD MANIFOLDS

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By virtue of Claim 3.6, we have (3.5) d(fn )∗ (µXn ) ((ψn ◦ ϕn )−1 (qn )) Z −1 = dNn ((ψn ◦ ϕn ) (qn ), fn (xn ))dµXn (xn ) (ϕn ◦fn )−1 (An ) Z −1 + dNn ((ψn ◦ ϕn ) (qn ), fn (xn ))dµXn (xn ) −1 Xn \(ϕn ◦fn ) (An ) Z ≤ dNn (zn , fn (xn ))dµXn (xn ) − bn (ϕn ◦ fn )∗ (µXn )(An ) (ϕn ◦fn )−1 (An ) Z −1 + dNn ((ψn ◦ ϕn ) (qn ), fn (xn ))dµXn (xn ) 

Xn \(ϕn ◦fn )−1 (An )

 κ en ≤ d(fn )∗ (µXn ) (zn ) − bn inf mXl − l∈N (diam Xn )pn Z + {dNn ((ψn ◦ ϕn )−1 (qn ), fn (xn )) − dNn (zn , fn (xn ))}dµXn (xn ) Xn \(ϕn ◦fn )−1 (An )

1 ≤ d(fn )∗ (µXn ) (zn ) − bn inf mXl + dNn ((ψn ◦ ϕn )−1 (qn ), zn )µXn (Xn \ (ϕn ◦ fn )−1 (An )) 2 l∈N for any sufficiently large n ∈ N. Since pn > 0, supl∈N pl < +∞, and inf l∈N diam Xl > 0, we obtain inf l∈N (diam Xl )pl > 0. Combining this with µXn (Xn \ (ϕn ◦ fn )−1 (An )) < κ en /(diam Xn )pn , limn→∞ κ en = 0, and limn→∞ (bn / dNn ((ψn ◦ ϕn )−1 (qn ), zn )) = 1, we get (3.6)

1 − bn inf mXl + dNn ((ψn ◦ ϕn )−1 (qn ), zn )µXn (Xn \ (ϕn ◦ fn )−1 (An )) 2 l∈N κ en dNn ((ψn ◦ ϕn )−1 (qn ), zn ) 1 < − dNn ((ψn ◦ ϕn )−1 (qn ), zn ) inf mXl + l∈N 4 inf l∈N (diam Xl )pl n 1 o κ en = dNn ((ψn ◦ ϕn )−1 (qn ), zn ) − inf mXl + <0 4 l∈N inf l∈N (diam Xl )pl

for any sufficiently large n ∈ N. Combining (3.5) with (3.6), we finally obtain −1 d(fn )∗ (µXn ) ((ψn ◦ ϕn ) (qn )) < d(fn )∗ (µXn ) (zn )

for any sufficiently large n ∈ N. Consequently we have a contradiction since the function d(fn )∗ (µXn ) attains its minimum at the point zn ∈ Nn . This completes the proof of the theorem.  Acknowledgements. The author would like to thank Professor Takashi Shioya for his valuable suggestions related to Theorem 1.2 and many discussions. He also thanks Professor Vitali Milman for useful comments. Without them, this work would have never been completed.

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