On convex perturbations with a bounded isotropic constant

Transcription

1 O covex perturbatios with a bouded isotropic costat B. Klartag Abstract Let K R be a covex body ad ε >. We prove the existece of aother covex body K R, whose Baach-Mazur distace from K is bouded by + ε, such that the isotropic costat of K is smaller tha c ε, where c > is a uiversal costat. As a applicatio of our result, we preset a slight improvemet o the best geeral upper boud for the isotropic costat, due to Bourgai. Itroductio Let K R be a covex body, i.e. a compact covex set with a o-empty iterior. We say that K is isotropic or that K is i isotropic positio, if V ol(k) =, the baryceter of K is at the origi, ad x ix jdx = L 2 Kδ i,j, () K for some umber L K >, where x = (x,..., x ) R are coordiates i R, ad δ i,j is Kroecker s delta. Whe K is isotropic, we say that L K as i () is the isotropic costat of K. It is well-kow (e.g., [2]) that for ay covex body K R, there exists a affie map T : R R such that T (K) is i isotropic positio. This affie map T is uique, up to left multiplicatio by a orthogoal trasformatio (e.g., [2]). We defie the isotropic costat of a arbitrary covex body K R to be L K := L T (K), where T : R R is ay affie map such that T (K) is i isotropic positio. The isotropic costat of K is well-defied, ad is ivariat uder affie trasformatios. See below for a more direct defiitio of the isotropic costat of a o-isotropic covex body. Amog all covex bodies i R, ellipsoids possess the miimal isotropic costat (this fact essetially goes back to Blaschke [3]. A proof appears, e.g., i [2]). It is straightforward to verify that c, the isotropic costat of a -dimesioal ellipsoid, satisfies c 2πe whe. Thus the miimal possible value of the isotropic costat of a covex body i R is well uderstood. I cotrast, it is ot eve kow what the order of The author is a Clay Research Fellow, ad was also supported by NSF grat #DMS

2 magitude of the maximal isotropic costat is, amog all covex bodies i R. This is related to a basic ope problem i asymptotic covex geometry, the validity of the hyperplae cojecture (e.g., [, 4, 2]). The hyperplae cojecture suggests that for ay covex body K R of volume oe, there exists a affie hyperplae H R such that where c > is a uiversal costat. V ol (K H) > c A equivalet formulatio of the hyperplae cojecture reads as follows: For ay dimesio, ad ay covex body K R, the isotropic costat L K is bouded from above by some uiversal costat (see [2] for the aforemetioed equivalece, ad for additioal equivalet, plausible, formulatios of the hyperplae cojecture). Furthermore, ay upper boud o the isotropic costat implies a lower boud o the volume of hyperplae sectios, as follows: For ay covex body K R of volume oe, there exists a hyperplae H R with V ol (K H) > c L K, where c > is a uiversal costat (see e.g., [2]). The hyperplae cojecture was verified for several large classes of covex sets: Ucoditioal covex bodies [4, 2], zooids, duals to zooids, [2] (see also [2]), bodies with a bouded outer volume ratio [2], radom bodies [8], uit balls of Schatte orms [9], ad others (e.g., [5]). A reductio of the problem to the case of bodies with a bouded volume ratio appears i [7, 8]. However, the best geeral boud kow to date is Bourgai s estimate [5], L K < c 4 log( + ) (2) for ay covex body K R. Bourgai s argumet formally deals oly with cetrally-symmetric sets. See [23] for the o-symmetric case, or the last remark i [7] for a reductio of the geeral problem to the case of cetrally-symmetric covex bodies. Additioal proofs of the boud (2) were preseted by Dar [] ad by Bourgai [6]. as For two covex bodies K, K 2 R, we defie their geometric distace d(k, K 2) = if { ab; a, b >, x, y R, } (K + x) K2 + y b(k + x). a Thus, the distace betwee K ad K 2 is small if, oce we apply suitable traslatios, the body K is close to a dilatio of the body K 2. Clearly, d(k, K 2) is ot larger tha the Baach-Mazur distace betwee K ad K 2 (see e.g., [3, page 767]). Our mai result is the followig theorem. Theorem. Let K R be a covex body, ad let ε >. The there exists a covex body T R such that. d(k, T ) < + ε. 2. L T < c ε. Here, c > is a uiversal costat. 2

3 A weaker versio of Theorem., with a logarithmic factor, was obtaied i [7]. A direct cosequece of the recet Paouris theorem [25, 26], is that if K, T R are covex bodies ad d(k, T ) < +, the L K ad L T have the same order of magitude. Thus, the case ε = i Theorem. etails the followig slight improvemet of (2). Corollary.2 Let K R be a covex body. The where c > is a uiversal costat. L K < c 4, The rest of the paper is orgaized as follows: I Sectio 2 we review some kow results related to log-cocave fuctios. Sectio 3 cotais a descriptio of our mai tool, a certai trasportatio of measure. Theorem. ad Corollary.2 are prove i Sectio 4. Throughout this paper, the letters c, C, c, c etc. deote positive uiversal costats, whose values are ot ecessarily the same i differet appearaces. We would like to emphasize that these costats are, i particular, idepedet of the dimesio. We use the otatio A B to abbreviate c A < B < c 2A, for c, c 2 >, uiversal costats. 2 Log-cocave fuctios I this sectio we summarize some facts, mostly stadard, o log-cocave fuctios. A fuctio f : R [, ) is log-cocave if for ay x, y R ad < λ <, f(λx + ( λ)y) f λ (x)f λ (y), (i.e., log f is cocave). A log-cocave fuctio is always measurable. A log-cocave fuctio f with < f < has momets of all orders. I particular its baryceter bar(f) = R xf(x)dx R f(x)dx R is well-defied, as well as its iertia matrix Cov(f) = (Cov(f) i,j) i,j=,...,, whose etries are x R ix jf(x)dx x Cov(f) i,j = R if(x)dx x R jf(x)dx f(x)dx f(x)dx f(x)dx R R R. We also refer to Cov(f) as the covariace matrix of f. For a log-cocave fuctio f : R [, ) with < f <, we defie its isotropic costat as ( ) supx R f(x) L f = (det Cov(f)) 2. (3) f(x)dx R It is straightforward to verify that L f = L f T for ay affie map T : R R, ad also that L f = L af for ay a >. We say that f is i isotropic 3

4 positio if sup x R f(x) = f(x)dx = ad Cov(f) is a scalar matrix. I this case, Cov(f) = L 2 f Id, where Id is the idetity matrix. We have already defied the isotropic costat of a covex body K R i Sectio. This defiitio is cosistet with (3) i the followig sese: Deote by K the characteristic fuctio of K, a log-cocave fuctio. The L K = L K. Let us describe yet aother characterizatio of the isotropic costat. We deote by ad, the stadard Euclidea orm ad scalar product i R, respectively. We also write S = {x R ; x = } for the uit sphere. Suppose that f : R [, ) is a log-cocave fuctio with < f <. The, as is prove i [2], L 2 f = if T :R R ( ) 2 supx R f(x) f(x)dx R T x 2 dx f(x), (4) R f(y)dy where the ifimum rus over all volume-preservig affie maps T : R R. The sigificace of log-cocave fuctios stems maily from the Bru- Mikowski type iequalities. Suppose that f : R [, ) is a logcocave fuctio. The, as follows from the Prékopa-Leidler iequality (e.g., first pages of [27]), for ay compact sets A, B R A+B 2 f(x)dx f(x)dx A B f(x)dx where A+B 2 = { x+y 2 ; x A, y B}. Cosequetly, log-cocave fuctios ejoy some cocetratio properties. For istace, Borell s lemma (e.g., [3, Page 77]) implies that for ay θ R ad p, x, θ f(x) dx ( x, θ p f(x) dx ) p < cp x, θ f(x) dx, R f R f R f (5) where c > is a uiversal costat. Aother immediate cosequece of Borell s lemma reads as follows: Let f : R [, ) be log-cocave with < f <, ad deote by M the media of the Euclidea orm with respect to f. That is, f(x)dx = f(x)dx. The by Borell s x <M 2 R lemma, x 2 f(x) dx M 2. (6) R f Next, we quote the results of K. Ball from []. The followig lemma is precisely the cotet of (6), (7) i []. Lemma 2. Suppose g, h, m : [, ) [, ) are three measurable fuctios, such that for ay r, s >, ( ) 2 m g(r) r+s s h(s) r+s r. (7) r + s 4

5 Let p, ad deote A = The, g(r)r p dr, B = S h(r)r p dr, S = 2 A + B. m(r)r p dr. The ext theorem is also due to K. Ball []. Sice the theorem is prove i [] oly for eve fuctios, for the reader s coveiece we sketch the straightforward adaptatio to the o-eve case below. Theorem 2.2 Let f : R [, ) be a log-cocave fuctio with f() >, ad let p. The the set { K p(f) = x R ; f(rx)r p dr f() } p is covex. Proof: The, A := Let x, y K p(f), ad deote g(r) = f(rx), h(r) = f(ry). g(r)r p dr f() p, B := h(r)r p dr f() p. We eed to show that x+y K 2 p(f). Equivaletly, if m(r) = f ( ) r x+y 2, the it is sufficiet to prove that Let r, s >. Set λ = to obtai ( ) 2rs m r + s S := m (r) r p dr f() p. s, u = rx, v = sy, ad use the log-cocavity of f r+s = f(λu + ( λ)v) f λ (u)f λ (v) = g(r) s r+s h(s) r r+s. Thus g, h, m satisfy requiremet (7) of Lemma 2.. From the coclusio of that lemma, S f(), ad the theorem follows. p The set K p(f) = {x R ; p f(rx)r p dr f()}, defied for ay Borel measurable fuctio f : R [, ), will play a importat rôle later o. Note that K p(f) for ay p, as f()r p dr = f(). Recall that for a set K R we deote by K the characteristic fuctio of K. Lemma 2.3 Let K R be a covex body cotaiig the origi. Let p. The, K p( K) = K. 5

6 Proof: For ay x R deote r x = sup{r ; rx K}, ad observe that p K(rx)r p dr = rx pr p dr = r p x. Thus, x K p( K) if ad oly if r x, which holds if ad oly if x K. Lemma 2.4 Let f, g : R [, ) be two measurable fuctios with f(x) f() = g() >, let p, ad deote m = sup. The, g(x)> g(x) K p(f) m p K p(g). (8) Proof: Suppose that x K p(f). The, ( ) g rm p x r p dr = mg(rx)r p dr f(rx)r p dr f() p = g() p. Therefore m p x K p(g) ad x m p K p(g). This proves (8). The ext lemma is due to Fradelizi [2, Theorem 4]. Lemma 2.5 Let g : R [, ) be a log-cocave fuctio such that < g <. Let x R = bar(g) be the baryceter of g. The, sup g(x) e g(x ). x R The followig lemma is a stadard oe-dimesioal computatio. It is almost idetical, e.g., to [7, Lemma 2.4]. For completeess, we sketch its easy roof. Lemma 2.6 Let be a iteger, ad let g : [, ) [, ) be a logcocave fuctio with g() =, < g(t)t dt < ad sup x g(x) e. The c < + e( + ) g(t)t dt ( g(t)t dt ) + where c, c 2 > are uiversal costats.! (( )!) + < c 2, (9) Proof: Set A = g(t)t dt, ad let r > be such that r e t dt = A. Sice g(t) e for ay t >, the each x > satisfies x g(t)t dt = A x g(t)t dt A mi{r,x} Cosequetly, by itegratig by parts we obtai g(t)t dt = x g(t)t dtdx r mi{r,x} e t dt = r e t dtdx = mi{r,x} r e t dt. e t dt = + (A) e( + ). 6

7 This proves the left had side of (9). Next we focus our attetio o the right had side of (9). Select a > such that e at t dt = g(t)t dt = A. () By (), it is impossible that always g(t) < e at or always g(t) > e at. Hece ecessarily t = if{t > ; e at g(t)} is fiite. The fuctio log g(t) log g is covex ad vaishes at zero, therefore g(t) = is odecreasig. Thus g(t) a for t < t, ad g(t) a for t > t. Equivaletly, t g(t) e at for t < t ad g(t) e at for t > t. We coclude that for x t, x g(t)t dt x e at e dt. () Usig () we deduce that () holds also for < x t. Thus () holds for all x >. By itegratig by parts, as before, we coclude that g(t)t dt = x g(t)t dtdx x e at t dtdx = To establish the right had side of (9), observe that t e at dt = ( ) + A ( + )! ad that (( )!) /. The proof is complete. ( )! Next we compare, alog the lies of [] ad [2], some volumetric characteristics of the fuctio f ad the body K +(f). e at t dt. Lemma 2.7 Let f : R [, ) be a log-cocave fuctio with < f <, ad suppose that its baryceter lies at the origi, i.e. xf(x) =. The also the baryceter of K +(f) lies at the origi. Furthermore, cl f < L K+ (f) < CL f where c, C > are uiversal costats. Proof: Accordig to Lemma 2.5, ecessarily f() >, sice otherwise f. Both the assumptios ad the coclusios of the lemma are ivariat uder replacemet of f by af, for ay a >. Thus we may assume that f() =. For θ S deote { } r θ = sup {t > ; tθ K +(f)} = sup t > ; ( + ) f(rtθ)r dr. From (2) we coclude that for ay θ S, r θ = ( ( + ) (2) ) f(rθ)r dr +. (3) 7

8 Itegratio i polar coordiates the yields x, θ dx (4) K + (f) = = S rθ ry, θ r drdy = y, θ ry + dy. + S f(ry) y, θ r drdy = x, θ f(x)dx =, S R sice the baryceter of f lies at the origi. We deduce from (4) that the baryceter of K +(f) lies at the origi. Furthermore, by arguig as i (4), we coclude that for ay θ S, x, θ dx = x, θ f(x)dx. (5) K + (f) R We itegrate by polar coordiates ad use (3) to obtai V ol(k +(f)) = r ( + ) θ dθ = S Accordig to Lemma 2.5, for ay x R, + S ( ) f(rθ)r + dr dθ. (6) f(x) e. (7) Based o (7), Lemma 2.6 implies that for ay θ S, ( ) f(rθ)r + dr f(rθ)r dr (8) (ote that the quatities i (8) are fiite; Sice < f <, the ay restrictio of f to a straight lie has a fiite itegral). Combiig (6) ad (8), we get V ol(k +(f)) f(rθ)r drdθ = f(x)dx. (9) S R Next, (5) ad (9) imply that for ay θ S, dx x, θ V ol(k x, θ f(x) dx. +(f)) R f K + (f) Usig (5), we deduce that for ay θ S, x, θ 2 dx K + (f) V ol(k x, θ 2 f(x) dx. (2) +(f)) R f The estimate (2) etails that the iertia matrices Cov(f) ad Cov(K +(f)) := Cov( K+ (f)) satisfy c Cov(f) < Cov(K +(f)) < c 2Cov(f) (2) 8

9 i the sese of positive defiite matrices, for some uiversal costats c, c 2 >. Accordig to (9), clearly V ol(k +(f)) ( f). Sice f() =, we coclude by (3), (7) ad (2) that L f L K+ (f). This sectio s results are cosolidated i the followig lemma. Lemma 2.8 Let K R be a covex body, ad let f : K (, ) be a log-cocave fuctio. Suppose that m > satisfies sup f(x) m if f(x). x K x K The there exist a covex set T R ad x R such that. (T x) K x m(t x). m 2. c L f < L T < c 2L f where c, c 2 > are uiversal costats. Proof: Suppose first that the baryceter of f lies at the origi. Multiplyig f by a positive costat, if ecessary, we may assume that f() =. Let T = K +(f) = { x R ; ( + ) f(rx)r dr f() }. The set T is covex accordig to Theorem 2.2. Accordig to our assumptios, sup K (x)> f(x) K(x) m m + K(x), sup f(x)> f(x) m m +. (22) Recall that K +( K) = K by Lemma 2.3. Lemma 2.4 ad (22) etail that m T K mt. Moreover, accordig to Lemma 2.7, the baryceter of T lies at the origi ad L T = L K+ (f) L f. Thus the lemma is prove, with x =, i the case where the baryceter of f is the origi. The geeral case is easily reduced to the case where xf(x)dx the baryceter of f lies at the origi. Ideed, set x = bar(f) = f(x)dx, ad cosider the log-cocave fuctio f(x) = f(x + x ), that is supported o K = K x. Sice the baryceter of f lies at the origi, we kow that T = K +( f) satisfies L T L f = L f, ad also m T K m T. Therefore T = T + x satisfies (T x) K x m (T x). m Sice L T = L T L f, the lemma is prove. 9

10 3 Trasportatio map Let K R be a covex body. We cosider the followig fuctio F K : R R, F K(x) = log e x,y dy K V ol(k). Our use of the fuctio F K is ispired by a remark by Gromov i [4]. The fuctio F K also resembles the partitio fuctios of statistical mechaics. It might be useful to ote that F K is defied, i priciple, o the dual space to R, ad that there is o eed to fix a scalar product i R i order to defie F K. A few simple properties of F K are established i the ext lemma. Lemma 3. Suppose K R is a covex body. The F K is C 2 -smooth, strictly covex, ad Im( F K) := { F K(x); x R } satisfies Im( F K) = it(k), the iterior of K. Furthermore, for ay x R deote by µ K,x the probability measure o R whose desity at y R equals e x,y K(y) K e x,z dz. The, for ay x R, F K(x) = bar(µ K,x) = y dµ K,x(y), R the baryceter of µ K,x. Additioally, [ ] [ ] Hess(F K)(x) = Cov(µ K,x) = y y dµ K,x(y) R y dµ K,x(y) R y dµ K,x(y) R, the covariace matrix of µ K,x. Here Hess stads for Hessia, ad x x stads for the matrix whose etries are (x ix j) i,j=,...,. Proof: The smoothess of F K is clear, as we are itegratig a smooth fuctio o a compact set. The strict covexity of F K follows from the Cauchy-Schwartz iequality, sice for ay x x 2 R, x +x 2,y dy e 2 V ol(k) < dy dy e x,y e V ol(k) x 2,y V ol(k). (23) K K ( Takig the logarithm of both sides i (23), we obtai that F x +x 2 ) K 2 < F (x )+F (x 2 ). Next, we differetiate uder the itegral sig to get that for 2 ay x R, K F K(x) = ye x,y dy K e x,y dy = y dµ K,x(y). (24) R K

11 Thus F K(x) is the baryceter of the measure µ K,x. Sice µ K,x is supported o the compact, covex set K, its baryceter bar(µ K,x) K. Therefore F K(x) K for ay x R. (25) Next, let y K be a extremal poit of K (i.e. there is o iterval cetered at y that is cotaied i K). There exists a supportig hyperplae for K, such that y is its oly cotact poit with K. Thus, there exist x R ad b R such that x, y = b, z K, z y x, z < b. Cosider the measure µ K,rx for large r >. Its desity is proportioal to z e r x,z K(z), ad it attais its uique maximum at y. Furthermore, it is straightforward to verify that as r, µ K,rx w δ y where δ y is the delta measure supported o y. Therefore, by (24), F K(rx) r y ad y Im( F K). Sice y was a arbitrary extremal poit of K, we coclude that Im( F K) cotais all extremal poits of K. Recall that Im( F K) is covex [4, Lemma 2.3], ad that K is the covex hull of its extremal poits. Therefore, K Im( F K). (26) Sice Im( F K) is ope (e.g., Lemma 2.2 i [4]), by combiig (25) ad (26) we coclude that Im( F K) is the iterior of K. This proves the first part of the lemma. It remais to compute the Hessia matrix of F K. Fix i, j. Differetiatio of (24) yields, 2 F K(x) x i x j = ad the lemma is prove. K yiyje x,y dy K e x,y dy K yie x,y dy K yje x,y dy (K e x,y dy ) 2 Suppose µ, µ 2 are two Borel measures o R, ad let T : R R be a measurable map. We say that T trasports µ to µ 2 if for ay Borel set A R, µ 2(A) = µ (T (A)). Equivaletly, for ay cotiuous, o-egative fuctio ϕ : R R, ϕ(x)dµ 2(x) = ϕ(t x)dµ (x). R R Lemma 3.2 Let F : R R be a strictly-covex, C 2 -smooth fuctio. Deote K = Im( F ), let λ K be the restrictio of the Lebesgue measure to K, ad defie µ to be the measure whose desity at x R equals = det HessF (x). dµ dx The F : R R trasports µ to λ K.

12 Proof: Let ϕ : R R be a cotiuous, o-egative fuctio. Sice F is strictly covex, the F : R R is oe-to-oe. Chagig variables x = F (y), we obtai ϕ(x)dx = ϕ( F (y)) det(hess(f (y)))dy = ϕ( F (y))dµ(y). Im( F ) R This completes the proof. Deote by µ K the measure o R whose desity at x is det Cov(µ K,x). Lemma 3. ad Lemma 3.2 tell us that F K trasports the measure µ K to the uiform measure o K. I particular, µ K(R ) = V ol(k). Thus, we may trasfer volumetric computatios o K to correspodig questios o the measure µ K. 4 Proof of the mai results Proof of Theorem.: By traslatig ad rescalig K, we may assume that V ol(k) = ad that the baryceter of K lies at the origi. I particular, cov(k, K) K K (27) where cov(a, B) deotes the covex hull of A ad B. By the Rogers- Shephard theorem [28], ( ) 2 V ol(k K) V ol(k) < 4. (28) Let K = [cov(k, K)], the polar body of cov(k, K). The K = {x R ; y K, x, y }. (29) Accordig to the Bourgai-Milma theorem [9], followed by (27) ad (28), V ol(k ) > c V ol(cov(k, K)) c > V ol(k K) > 4c. (3) Next, Recall the defiitio of the measure µ K from Sectio 3. That is, for ay x R, we defie a probability measure µ K,x whose desity at y R equals e x,y K(y) K e x,z dz. The, we defie µ K to be the measure whose desity at x equals det Cov(µ K,x) = det Hess(F K)(x). By Lemma 3., Im( F K) is the iterior of K. Accordig to Lemma 3.2, there exists a map that trasports the measure µ K to the uiform measure o K. I particular, Thus, V ol(εk ) µ K(εK ) < µ K(R ) = V ol(k) =. mi det Cov(µK,x) det Cov(µ K,x)dx = µ K(εK ) <. εk x εk (3) 2

13 Accordig to (3) ad (3), Let x εk be such that mi det Cov(µK,x) < x εk det Cov(µ K,x) < ( ) C. ε ( ) C. (32) ε The measure µ K,x is log-cocave; Ideed, its desity is proportioal to f(y) := e x,y K(y), which is the product of e x,y ad K(y), both logcocave. Also, by the defiitio of the isotropic costat (3), ( det Cov(µ K,x) = R f(y)dy sup y R f(y) Sice x εk ad f(y) = e x,y K(y), the by (29), ) 2 L 2 f. (33) sup f(y) = sup e x,y e ε. (34) y R y K Also, by Jese s iequality, ( ) f(y)dy = e x,y dy exp x, y dy =. (35) R K K Now (32), (33), (34) ad (35) imply that L 2 f < e 2ε ( C ε ) ad hece L f < c ε. (36) The fuctio f : K [, ) is log-cocave, ad e ε if f(y) sup f(y) e ε. (37) y K y K We may ivoke Lemma 2.8, based o the estimate (37). By the coclusio of that lemma there exists a covex set T R, with L T L f such that d(k, T ) < e ε + eε, ( < ε < ). However, by (36) we kow that L T < cl f < C ε. This completes the proof. Next we prove Corollary.2. We begi by quotig Paouris theorem [25, 26]. Theorem 4. (Paouris) Let K R be a isotropic covex body. The for ay t >, V ol(k \ ct L KD) < e t, where D = {x R ; x } is the uit Euclidea ball, ad c > is a uiversal costat. Our ext lemma is a cosequece of Theorem 4.. 3

14 Lemma 4.2 Let K, T R be covex bodies, ad t. Suppose that The, d(k, T ) < + where c > is a uiversal costat. L T < ctl K, t. (38) Proof: We may assume that t <, as otherwise the coclusio of the lemma is trivial, sice it is easy to prove that L T < c. (For example, if V ol(t ) =, the there exists a directio i which the width of T is smaller tha c, ad thus there exists a hyperplae sectio whose volume is larger tha with c + t (K + x ) (T + y ) ). Accordig to (38) there exist x, y R ( + t ) (K + x ). (39) Applyig a affie trasformatio to both K ad T, we may suppose that V ol(k) =, that the baryceter of K is at the origi, ad that K is isotropic. Let us set T = (T + y + ) x. By (39), T K. t Additioally, agai from (39), V ol( T ) = ( ) V ol(t ) ( ) + t 2 V ol(k) > e 2t. (4) + t Accordig to Theorem 4., we kow that V ol(k \ ct L KD) < e 4t (4) for some uiversal costat c >. Sice T K, the (4) ad (4) imply that V ol( T ct L KD) 2 V ol( T ). Therefore, the media of the fuctio x x o T, with respect to the uiform measure o T, is ot larger tha ct L K. Sice T is covex, by (6), T x 2 dx V ol( T < Ct L K (42) ) for some uiversal costat C >. Accordig to (4) ad (4), The lemma is prove. L T = L T = L T C tlk V ol( T ) < c tl K. Proof of Corollary.2: Let K R be a covex body, ad let us set ε =. Accordig to Theorem., there exists a covex body T R with d(k, T ) < + ε = + (43) 4

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