We prove that in every finite dimensional normed space, for “most” pairs (x, y) of points in the unit ball, ║x − y║ is more than √2(1 − ε). As a consequence, we obtain a result proved by Bourgain, using QS-decomposition, that guarantees an exponentially large number of points in the unit ball any two of which are separated by more than √2(1 − ε)

Ball, KM

Villa, R

English

UCL Discovery

CONCENTRATION OF THE DISTANCE INFINITE DIMENSIONAL NORMED SPACESJUAN ARIAS-DE-REYNA, KEITH BALL AND RAFAEL VILLAAbstract. We prove that in every finite dimensional normed space, for"most" pairs (x, y) of points in the unit ball, ||x —yll is more than yj2(\ - s).As a consequence, we obtain a result proved by Bourgain, using QS-decomposi-tion, that guarantees an exponentially large number of points in the unit ballany two of which are separated by more than ^2(1 - £).Introduction and previous results. Let X= (R", ||. . . ]|) be an n-dimensionalnormed space, K={xeW: ||x||<l} its closed unit ball and let m be theLebesgue measure restricted to K, normalized so that m(K) = 1. It is easy tosee that when n is large, most of the points in K lie near its surface: so theirnorm is about 1. In this article, our aim is to investigate the typical behaviourof the distance between two points in the ball.The investigation is motivated by a number of recent results showing thatin a wide variety of special spaces, it is possible to find many points, any twoof which are roughly distance 1 apart. Given X of dimension n, let N= N(e)be the highest cardinality of a subset T of X such that1-£<||JC-J>K1 + £, for all x, ye T, x#y. (1)A volume argument easily shows that AT(e)^exp {(p(e)n} for some functionq>, independent of X. In [BBK] an exponential estimate from below is obtained,for finite dimensional spaces with a 1-subsymmetric basis: in particular, forexample, for lp spaces. Some extensions of this result appear in [BB], wheresharp estimates are given for the case of the space lp(l™) (1 ^p, q< oo). Relatedresults can be found in [BPS1], [BPS2] and [BMW].The results in each of these articles depend upon the construction of someprobability measure on the unit ball with respect to which, two independentrandom vectors are almost exactly a fixed distance apart with very high prob-ability. In this article we consider the simplest such measure, Lebesgue measure,and examine the distribution of the random variable ||x —y|| (x,yeK),F(t)=m<S)m{(x,y)eKxK: \\x-y\\^t}.If F(t) jumps from 5 to 1 - 5 as t goes from a(\ - e) to a{\ + e), then we canfind N points x,, .. . , xNeK, such that a~xx\,. . . , axxN satisfy (1), as long as2N2S < 1. Our interest is in exponentially large sets of points so we shall lookfor a number a for which we have something likeF(a(\ + e))-F(a(\ - e)) > 1 -exp {-2ce2n}for some constant c.[MATHEMATIKA, 45 (1998), 245-252]246 J. ARIAS-DE-REYNA, K. BALL AND R. VILLAIn Section 1, we show that there are normed spaces for which there is nothreshold behaviour for the distribution function. However, our main resultguarantees that in all normed spaces, the distance between "most" pairs ofpoints in the unit ball is greater than ^2(1 - e). We show, in fact, that for anyspace,Hence, in particular, if the distance does concentrate around a number, thenthis number must be no smaller than ^/2. This result is sharp in that, forEuclidean space, there is a threshold at ,/2. (In this caseF(V2(1 - e)) >-L (1 - £2(2- e)2r\2Vso our argument actually recovers the "correct" exponent.)Section 2 is concerned with spaces which do exhibit some kind of thresholdbehaviour. It is shown in [GM] (see [A] for an easier proof along the samelines) that the balls of uniformly convex spaces enjoy a concentration ofmeasure phenomenon: if AT is such a set, then for any e > 0 and any Borel setAcK with m(A)^l/2, the measure of the expanded set AE ={xeK: dist (x, A) ^ e) satisfiesfor some positive function cp. We have included a very short proof of this fact.We then observe that such a concentration of measure guarantees thresholdbehaviour for our distribution F.We should mention that there are some results known for isometricembeddings, see [P], and that infinite-dimensional analogues of some of theseproblems appear in [D] and [EO].§1. Large distances. As above, let X=(U", ||. . . ||) be an «-dimensionalnormed space, let K= {xeU": \\x\\ ^ 1} be its closed unit ball and let m be theLebesgue measure on A"normalized so that m(K) = 1. The distribution functionof the distance between two independent points, each distributed uniformly onK, is given by,= m®m{{x,y)eK*K: \\x-y\\^t}.It is easy to check thatF(t)= m(K n (x + K))dm(x).We begin the section with some simple examples showing, among other things,that there are spaces for which F does not have a sharp threshold.CONCENTRATION OF DISTANCE IN NORMED SPACES 247Examples. (1) X = l2. It is clear from standard results that the Eucli-dean distance between independent random points in the Euclidean ball, con-centrates around >/2 since both points are very close to the surface and eachone is almost certain to be near the equator "perpendicular to" the other.However, it will be useful later, if we examine the distribution more carefully.For any xe2K,im(Kn(x + K)) = —vn-t (l-s2)u"~l)ds,vn Jr/2where r= \\x\\ and vn is the volume of the unit ball in l\. Integrating over tKwe getF(t)=2nl^ivn0From this we get that the density of the random variable \\x — y\\ is roughly/K\ 4(A similar expression will reappear later on.)(2) X = l"x.F{t)=pr"is «-th power of the two-dimensional measure of the setand so,Since t(A—i) attains its maximum at t = 2, this shows that the distance in theball of l"t concentrates around a = 2.(3) The next example shows that there are normed spaces where concentra-tion of the distance does not occur. The real point is that concentration canoccur at different points for different spaces. Let X=(l"2® U)o0 = (U"+l,||... ||), where!!(A-,>')II=SUP{||X||2,|>'|}, xeln2,yeU.Thus AT is a "circular" cylinder in U"+].It is easy to check that for a Cartesian product set, the distribution functionF is just the product of the distributions for the two factors. So for our space,F(t)=F] (t)F2(t), F\ being the distribution function for l"2 and F2 being the onefor U. F, has a sharp threshold at -Jl and so is almost equal to 1 on the248 J. ARIAS-DE-REYNA, K. BALL AND R. VILLAinterval [^2(1 + e), 2]. Thus, F is almost the same as F2 from ~Jl onwards.From example (2) we see that F2 is given byand hence F has no sharp threshold.Despite examples like the preceding one, our main result, Theorem 1 below,shows that in any normed space "most" of the pairs of points in the unit ballare separated by more than J2( 1 - e): in particular, that we may find exponen-tially many, that are so separated. This fact was proved by Bourgain usingQS-decomposition (see [FL] for a proof). It was proved in [EO] that in anyinfinite-dimensional normed space, there is an infinite subset of the unit ballwhose elements are (1 + e)-separated, for some £>0 depending on the normedspace. Theorem 1 seems stronger than this, in spirit, but in [BRR] it wasshown that the greatest separation of any infinite subset in the unit ball of lp(1 </?< oo) is 2]/p: so Theorem 1 cannot be used to improve the infinite dimen-sional statement.THEOREM 1. Let X be an n-dimensional normed space, K its closed unitball and m, the Lebesgue measure on K normalized so that m(K)= 1. Then, foranyQ<t<j2m®m{(x,y)eKxK: \\x-y\\^tIf t = s]2( 1 — £) the latter is no more thanexp { — s2n/2}.Note. The sharpness of the result can be gauged by referring to theexample X=l" discussed earlier.Proof. For any te[O, 2], let= m<g)m{(x, y)eK* K: \\x-y\\^t}.It is clear that F(t) is a value of the threefold convolution of the characteristicfunctions of K (twice) and tK,To estimate this we use the sharp form of Young's inequality for convolutionson W (see note below): for any p, q, r^\ with1 1 1-+-+-=2Parthere is a constant C(p, q, r; n)>0 such that for any functions feLp(W),geLq(W) and heL\W), we have\f*g*h(O)\^C(p,q,r;nCONCENTRATION OF DISTANCE IN NORMED SPACES 249The sharp constant C(p,q,r;n) equals (CpCqCr)n, where C2=pl/pp' x/p(// being the conjugate exponent to p). If r = q we obtainC(p,q,q;n) ={2q) 2«- l/2-ifor any p,2 \qwith (\/p) + (2/q) = 2. Hence, for any q> 1,L 2For t<sf2, the last expression is minimum for q- ^(4-t2) and for this q theexpression simplifies to giveThen, if 0 < e < 1, we haveY(V2(l-£))2(4-(V2(lV ~ 4{-e2«/2}.= (l-£2(2-£)2r/2Note. The sharp constant in Young's inequality was determined by Beck-ner, [Be], in connection with his study of the Fourier transform. An importantextension of Young's inequality, again with sharp constant, was proved byBrascamp and Lieb, [BL]. Recently, a very elegant new proof of their inequal-ity was found by F. Barthe [Ba].§2. A concentration of measure phenomenon. Uniformly convex spaces. Webegin this section with a short proof of the concentration phenomenon inuniformly convex spaces, proved by Gromov and Milman, [GM]. The originof our argument is in a paper of Talagrand [T] concerning Gauss space. Asimplification of his argument was found by Maurey [M], and was used bySchmuckenslager [S] to study uniformly convex spaces. The proof below is inthe same spirit. Unfortunately, the proof is now so short that one cannot seethe ideas behind it.For a uniformly convex normed space X, we define the modulus of convex-ity, 8, of X byHl2\\x-y\\>e,\\x\\^l,\\y\\^lThis definition is slightly different from that in the standard texts but thedifference is of no real consequence. As before, for a subset A of X and a250 J. ARIAS-DE-REYNA, K. BALL AND R. VILLAnumber £>0 we write Ac for the set of points whose distance (in norm) fromA is at most s.THEOREM 2. Let X be an n-dimensional uniformly convex normed space,K its closed unit ball, m the normalized Lebesgue measure on K and let 8 be themodulus of convexity of X. Then for any Borel set AaK with m(A) ^ 2 and any£>0,m(Al)>\-2e~2nS(':).Proof. Recall the multiplicative Brunn-Minkowski inequality:2Suppose A c K and putB={yeK: d(y,If xeA and yeB then |jx-j|| ^e and hencex + yThereforeA + Band soThe remainder of this section is devoted to showing that a concentrationof measure such as the above, suffices to guarantee that the distance betweentwo independent points in the unit ball, concentrates around some number.THEOREM 3. Let X be an n-dimensional normed space, K its closed unitball and m the normalized Lebesgue measure on K. Assume that there exists anincreasing function <p such that for any e > 0 and any Borel set AczK satisfyingThen there exists a number ae[\, 2] such that for any e>0,m®m{(x, y)eK* K: a(l - e) <\\x - y\\ ^a(l + e)} ^z I -4exp {-and hence, for any £>0 there exist Npoints x\, . . . , xN&X so that\- £^\\Xi~Xj\\^\ + £as long as N<(l/2) exp {(p(e/6)n/2}.CONCENTRATION OF DISTANCE IN NORMED SPACES 251Proof. For each xeK, consider the function fx : AT-> R defined by fx{y) =||x— v||. Let a(x) be the median offx so that the ball of radius a(x) includeshalf of K. (The median is clearly unique.) Whatever the dimension, \K hasvolume at most one half that of K so a(x) is at least \ for each x.Let a be a median of the function a. We have a^-\.Now let A = {xeK: a(x)^a}. Since AaE c{xeK: a(x)^a(l + e)}, the con-centration hypothesis ensures that,m({xeK: a(x)^a(\ + £)})^l-exp {-(p(ae)n} $s 1 -exp {-(p(s/2)n}.Similarlym({xeK: a(x)^a(\ - e)} )^ 1 -exp {and henceThe same argument can be applied for each fixed xeK to getm({yeK: a(x)(\- E)^\\x-y\\^a(x)(l + s)})^\-2exp {-cp(£/2)n}.These two inequalities givem®m({{x, y)eKxK:a(\- s)2^ \\x-y\\ <a(l + e)2} )^ 1 -4 exp {~^(e/2)«},and thus, for any s < 1m®m({(x,y)eKxK:a(\-3e)^\\x-y\\^a(\+3£)})> 1-4exp {-(p{e/2)n).Som®m({(x,y)eKx K:a(\ - e)^ \\x-y\\ <a(l + £)})> 1-4exp {-?)(e/6)«},for every e>0, as required.The conclusion concerning the choice of N points is now obvious.Acknowledgements. J.A. de R. and R.V. thank the Spanish DGICYT(PB96-1327) for grants. K.B. thanks the U.S. NSF for a grant. This paperincludes part of the Ph.D. thesis of R.V.ReferencesA. S. Alesker. On the Gromov Milman theorem regarding the concentration phenomenonon uniformly convex sphere. To appear.Ba. F. Barthe. Geometric and Functional Analysis. To appear.BB. J. Bastero and J. Bernues. Applications of deviation inequalities on finite metric sets.Math. Nachr.. 153 (1991), 33 41.BBK. J. Bastero, J. Bernues and N. Kalton. Embedding /"^-cubes in finite dimensional1-subsymmetric spaces. Rev. Mat. Univ. Complut. Madrid, 2 (1989), 47 52.BPS1. J. Bastero, A. Pefia and G. Schechtman. Embeddings /^-cubes in the orbit of an elementin commutative and noncommutative /^-spaces. Seminario Dpto. Andlisis Matemat-ico, Univ. Complut. Madrid.BPS2. J. Bastero, A. Pefia and G. Schechtman. Embeddings lnx,-cubes in low dimensionalSchatten classes. Geometric Aspects of Functional Analysis (Israel, 1992-1994),252 CONCENTRATION OF DISTANCE IN NORMED SPACES(1995), 5-11.Be. W. Beckner. Inequalities in Fourier analysis. Ann. of Math., 102 (1975), 159-182.BMW. J. Bourgain, V. Milman and H. Wolfson. On type of metric spaces. Trans. Amer. Math.Soc, 294 (1986), 295-317.BL. H. J. Brascamp and E. H. Leib. Best constants in Young's inequality, its converse, andits generalization to more than three functions. Advances in Math., 20 (1976), 151173.BRR. J. A. C. Burlak, R. A. Rankin and A. P. Robertson. The packing of spheres in thespace /,. Proc. Glasgow Math. Assoc, 4 (1958), 22-25.D. T. Dominguez-Benavides. Some properties of the set and ball measures of noncompact-ness and applications. J. London Math. Soc. (2), 34 (1986), 120-128.EO. J. Elton and E. Odell. The unit ball of every infinite-dimensional normed linear spacecontains a(l + £)-separated sequence. Colloq. Math., 44 (1981), 105 109.FL. Z. Fiiredi and P. A. Loeb. On the best constant for the Besicovitch covering theorem.Proc. Amer. Math. Soc, 121 (1994), 1063 1073.GM. M. Gromov and V. D. Milman. Generalization of the spherical isoperimetric inequalityto uniformly convex Banach spaces. Compositio Math., 62 (1987), 263 282.K. C. A. Kottman. Subsets of the unit ball that are separated by more than one. StudiaMath., 53 (1975), 15 27.M. B. Maurey. Some deviation inequalities. Geom. Fund. Anal., 1 (1991), 188 197.Pe. C. M. Petty. Equilateral sets in Minkowski spaces. Proc. Amer. Math. Soc, 29 (1971).369-374.S. M. Schmuckenschlager. A concentration of measure phenomenon on uniformly convexbodies. Geometric Aspects of Functional Analysis (Israel, 1992 1994), (1995), 275-287.T. M. Talagrand. A new isoperimetric inequality and the concentration of measure phe-nomenon. Geometric Aspects of Functional Analysis (Israel Seminar, 1989-1990).Lecture Notes in Math., 1469 (1991), 94-124.Keith M. Ball,Department of Mathematics,University College London,Gower Street,London WCIE 6BT.e-mail: kmb@math.ucl.ac.ukJuan Arias-de-Reyna,Depto. Analisis Matematico,Facultad de Matematicas,Universidad de Sevilla,c/Tarfia, S/N,41012 Sevilla,Spain.e-mail: arias@cica.esRafael Villa,Depto. Analisis Matematico,Facultad de Matematicas,Universidad de Sevilla,c/Tarfia, S/N,41012 Sevilla,Spain.e-mail: rvcaro@cica.es46B04: FUNCTIONAL ANA LYSIS; Normedlinear spaces, and Banach spaces, Ban-ach lattices; Isometric theory of Banachspaces.Received on the 2nd of December, 1997.

A concentration of measure phenomenon on uniformly convex bodies. Geometric Aspects of Functional Analysis (Israel,

A new isoperimetric inequality and the concentration of measure phenomenon. Geometric Aspects of Functional Analysis (Israel Seminar,

Best constants in Young's inequality, its converse, and its generalization to more than three functions.

Embedding /&quot;^-cubes in finite dimensional 1-subsymmetric spaces.

Embeddings l n x,-cubes in low dimensional Schatten classes. Geometric Aspects of Functional Analysis (Israel,

Equilateral sets in Minkowski spaces.

Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces.

Inequalities in Fourier analysis.

London, Gower Street, London WCIE 6BT. e-mail: kmb@math.ucl.ac.uk Juan Arias-de-Reyna, Depto. Analisis Matematico, Facultad de Matematicas, Universidad de

On type of metric spaces.

Received on the 2nd of December,

Some deviation inequalities.

Some properties of the set and ball measures of noncompactness and applications.

Subsets of the unit ball that are separated by more than one.

The packing of spheres in the space /,.

The unit ball of every infinite-dimensional normed linear space contains a(l + £)-separated sequence.

Concentration of the distance between points in the unit ball

Concentration of the distance between points in the unit ball

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