On the distribution of the ψ 2 -norm of linear functionals on isotropic convex bodies

Transcription

1 O he disribuio of he ψ 2 -orm of liear fucioals o isoropic covex bodies A. Giaopoulos, G. Paouris ad P. Valeas Absrac I is ow ha every isoropic covex body K i R has a subgaussia direcio wih cosa r = O( log ). This follows from he upper boud Ψ 2(K) 1/ c log L K for he volume of he body Ψ 2(K) wih suppor,θ fucio h Ψ2 (K)(θ) := sup q 2 q q. The approach i all he relaed wors does o provide esimaes o he measure of direcios saisfyig a ψ 2-esimae wih a give cosa r. We iroduce he fucio ψ K() := σ ( {θ S 1 : h Ψ2 (K)(θ) c log L ) K} ad we discuss lower bouds for ψ K(), 1. Iformaio o he disribuio of he ψ 2-orm of liear fucioals is closely relaed o he problem of boudig from above he mea widh of isoropic covex bodies. 1 Iroducio A covex body K i R is called isoropic if i has volume 1, i is ceered (i.e. i has is ceer of mass a he origi), ad here exiss a cosa L K > 0 such ha (1.1) x, θ 2 dx = L 2 K K for every θ S 1. I is ow (see [19]) ha for every covex body K i R here exiss a iverible affie rasformaio T such ha T (K) is isoropic. Moreover, his isoropic posiio of K is uiquely deermied up o orhogoal rasformaios; herefore, if we defie L K = L K where K is a isoropic affie image of K, he L K is well defied for he affie class of K. A ceral quesio i asympoic covex geomery ass if here exiss a absolue cosa C > 0 such ha L K C for every covex body K. Bourgai [4] proved ha L K c 4 log for every symmeric covex body K i R. The bes ow geeral esimae is currely L K c 4 ; his was proved by Klarag i [10] see also [12]. 1

2 Le K be a ceered covex body of volume 1 i R. We say ha θ S 1 is a subgaussia direcio for K wih cosa r > 0 if, θ ψ2 r, θ 2, where { } (1.2) f ψα = if > 0 : exp (( f(x) /) α ) dx 2, α [1, 2]. K V. Milma ased if every ceered covex body K has a leas oe subgaussia direcio (wih cosa r = O(1)). By he formulaio of he problem, i is clear ha oe ca wor wihi he class of isoropic covex bodies. Affirmaive aswers have bee give i some special cases. Bobov ad Nazarov (see [2] ad [3]) proved ha if K is a isoropic 1 ucodiioal covex body, he, θ ψ2 c θ for every θ S 1 ; a direc cosequece is ha he diagoal direcio is a subgaussia direcio wih cosa O(1). I [23] i is proved ha every zooid has a subgaussia direcio wih a uiformly bouded cosa. Aoher parial resul was obaied i [24]: if K is isoropic ad K (γ L K )B2 for some γ > 0, he (1.3) σ ( θ S 1 :, θ ψ2 c 1 γl K ) exp( c2 2 /γ) for every 1, where σ is he roaioally ivaria probabiliy measure o S 1 ad c 1, c 2 > 0 are absolue cosas. The firs geeral aswer o he quesio was give by Klarag who proved i [11] ha every isoropic covex body K i R has a subgaussia direcio wih a cosa which is logarihmic i he dimesio. A aleraive proof wih a slighly beer esimae was give i [6]. The bes ow esimae, which appears i [7], follows from a upper boud for he volume of he body Ψ 2 (K) wih suppor fucio, θ q (1.4) h Ψ2(K)(θ) := sup. 2 q q,θ I is ow ha, θ ψ2 sup q 2 q q, ad hece, h Ψ2(K)(θ), θ ψ2. The mai resul i [7] saes ha (1.5) c 1 L K Ψ 2 (K) 1/ c 2 log L K, where c 1, c 2 > 0 are absolue cosas. A direc cosequece of he righ had side iequaliy i (1.5) is he exisece of subgaussia direcios for K wih cosa r = O( log ). Wih a small amou of exra wor, oe ca also show ha if K is a ceered covex body of volume 1 i R, he here exiss θ S 1 such ha (1.6) {x K : x, θ c, θ 2 } e 2 log (+1) for all 1, where c > 0 is a absolue cosa. The approach i [11], [6] ad [7] does o provide esimaes o he measure of direcios for which a isoropic covex body saisfies a ψ 2 -esimae wih a give 2

3 cosa r. Klarag obais some iformaio o his quesio, bu for a differe posiio of K. More precisely, i [11] he proves ha if K is a ceered covex body of volume 1 i R he, here exiss T SL() such ha he body K 1 = T (K) has he followig propery: here exiss A S 1 wih measure σ(a) 4 5 such ha, for every θ A ad every 1, (1.7) {x K 1 : x, θ c, θ 2 } e c 2 log 2 log 5 (+1) I his resul, K 1 is he l-posiio of K (his is he posiio of he body which esseially miimizes is mea widh; see [27]). The firs aim of his oe is o pose he problem of he disribuio of he ψ 2 -orm of liear fucioals o isoropic covex bodies ad o provide some firs measure esimaes. To his ed, we iroduce he fucio ( (1.8) ψ K () := σ {θ S 1 : h Ψ2(K)(θ) c ) log L K }. The problem is o give lower bouds for ψ K (), 1. We prese a geeral esimae i Secio 4: Theorem 1.1. Le K be a isoropic covex body i R. For every 1 we have (1.9) ψ K () exp( c/ 2 ), where c > 0 is a absolue cosa. For he proof of Theorem 1.1 we firs obai, for every 1, some iformaio o he ψ 2 -behavior of direcios i a arbirary -dimesioal subspace of R : Theorem 1.2. Le K be a isoropic covex body i R. (i) For every log 2 / log ad every F G, here exiss θ S F ha such (1.10), θ ψ2 C / L K, (ii) For every 1 log 2 ad every F G, here exiss θ S F such ha (1.11), θ ψ2 C / log 2 L K, (iii) For every / log ad every F G, here exiss θ S F such ha (1.12), θ ψ2 C log L K, where C > 0 is a absolue cosa. I is ow (for example, see [14]) ha every isoropic covex body K is coaied i [( + 1)L K ]B 2. This implies ha he ψ 2 -orm is Lipschiz wih cosa 3

4 O( L K ). The, Theorem 1.2 is combied wih a simple argume which is based o he fac ha he ψ 2 -orm is sable o a spherical cap of he appropriae radius. Noe ha ψ K () = 1 if c / log. Therefore, he boud of Theorem 1.1 is of some ieres oly whe 1 c / log. Acually, if c 4 he we have much beer iformaio. I Secio 5 we give some esimaes o he mea widh of he L q ceroid bodies of K ad of Ψ 2 (K); as a cosequece, we ge: Proposiio 1.3. Le K be a isoropic covex body i R. c 4 1 / log oe has For every (1.13) ψ K () 1 e c22 log, where c 1, c 2 > 0 are absolue cosas. Deeper udersadig of he fucio ψ K () would have impora applicaios. The sregh of he available iformaio ca be measured o he problem of boudig from above he mea widh of isoropic covex bodies. From he iclusio K [( + 1)L K ]B 2, oe has he obvious boud w(k) cl K. However, a beer esimae is always possible: for every isoropic covex body K i R oe has (1.14) w(k) c 3/4 L K, where c > 0 is a absolue cosa. There are several approaches ha lead o he esimae (1.14). The firs oe appeared i he PhD Thesis of M. Harzoulai [9] ad was based o a resul from [5] regardig he mea widh of a covex body uder assumpios o he regulariy of is coverig umbers. The secod oe is more rece ad is due o P. Pivovarov [28]; i relaes he quesio o he geomery of radom polyopes wih verices idepedely ad uiformly disribued i K ad maes use of he coceraio iequaliy of [25]. A hird very direc proof of his boud ca be based o he heory of L q ceroid bodies which was developed by he secod amed auhor (see Secio 5). I Secio 6 we propose oe more approach, which ca exploi our owledge o ψ K (). 2 Bacgroud maerial 2.1. We wor i R, which is equipped wih a Euclidea srucure,. We deoe by 2 he correspodig Euclidea orm, ad wrie B 2 for he Euclidea ui ball, ad S 1 for he ui sphere. Volume is deoed by. We wrie ω for he volume of B 2 ad σ for he roaioally ivaria probabiliy measure o S 1. The Grassma maifold G, of -dimesioal subspaces of R is equipped wih he Haar probabiliy measure µ,. Le ad F G,. We will deoe by P F he orhogoal projecio from R oo F. We also defie B F := B 2 F ad S F := S 1 F. The leers c, c, c 1, c 2 ec. deoe absolue posiive cosas which may chage from lie o lie. Wheever we wrie a b, we mea ha here exis absolue 4

5 cosas c 1, c 2 > 0 such ha c 1 a b c 2 a. Also if K, L R we will wrie K L if here exis absolue cosas c 1, c 2 > 0 such ha c 1 K L c 2 K A covex body i R is a compac covex subse C of R wih o-empy ierior. We say ha C is symmeric if x C implies ha x C. We say ha C is ceered if i has ceer of mass a he origi, i.e. x, θ dx = 0 for every C θ S 1. The suppor fucio of a covex body C is defied by (2.1) h C (y) = max{ x, y : x C}, ad he mea widh of C is (2.2) w(c) = h C (θ)σ(dθ). S 1 For each < p <, p 0, we defie he p-mea widh of C by ( ) 1/p (2.3) w p (C) = h p C (θ)σ(dθ). S 1 The radius of C is he quaiy R(C) = max{ x 2 : x C} ad, if he origi is a ierior poi of C, he polar body C of C is (2.4) C := {y R : x, y 1 for all x C}. A ceered covex body C is called almos isoropic if C has volume oe ad C T (C) where T (C) is a isoropic liear rasformaio of C. Fially, we wrie C for he homoheic image of volume 1 of a covex body C R, i.e. C := C C 1/ Le K be a covex body of volume 1 i R. For every q 1 ad y R we defie ( 1/q (2.5) h Zq(K)(y) := x, y dx) q. K We defie he L q -ceroid body Z q (K) of K o be he cerally symmeric covex se wih suppor fucio h Zq(K). Noe ha K is isoropic if ad oly if Z 2 (K) = L K B 2. I is clear ha Z 1 (K) Z p (K) Z q (K) Z (K) for every 1 p q, where Z (K) = cov{k, K}. If T SL() he Z p (T (K)) = T (Z p (K)). Moreover, as a cosequece of Borell s lemma (see [20, Appedix III]), oe ca chec ha (2.6) Z q (K) cq Z 2 (K) for every q 2 ad, more geerally, (2.7) Z q (K) c q p Z p(k) 5

6 for all 1 p < q, where c 1 is a absolue cosa. Also, if K is ceered, he (2.8) Z q (K) c 1 K for all q, where c 1 > 0 is a absolue cosa Le C be a symmeric covex body i R. We wrie C for he orm iduced o R by C. We also defie (C) as he larges posiive ieger for which he measure of F G, for which 1 2 w(c)b F P F (C) 2w(C)B F is greaer ha +. The parameer (C) is deermied, up o a absolue cosa, by he mea widh ad he radius of C: There exis c 1, c 2 > 0 such ha (2.9) c 1 w(c)2 R(C) 2 (C) c 2 w(c)2 R(C) 2 for every symmeric covex body C i R. The lower boud follows from Milma s proof of Dvorezy s heorem (see [18]) ad he upper boud was proved i [21]. The q-mea widh w q (C) is equivale o w(c) as log as q (C). As Liva, Milma ad Schechma prove i [16], here exis c 1, c 2, c 3 > 0 such ha for every symmeric covex body C i R we have: 1. If 1 q (C) he w(c) w q (C) c 1 w(c). 2. If (C) q he c 2 q/ R(C) wq (C) c 3 q/ R(C) For every q >, q 0, we defie he quaiies I q (K) by ( (2.10) I q (K) := K x q 2 dx ) 1/q. I [26] ad [25] i is proved ha for every 1 q /2, (2.11) I q (K) /q w q (Z q (K)) ad (2.12) I q (K) /q w q (Z q (K)). We defie (2.13) q (K) := max{ : (Z (K)) }. The, he mai resul of [26] saes ha, for every ceered covex body K of volume 1 i R, oe has (2.14) I q (K) I q (K) for every 1 q q (K). I paricular, for all q q (K) oe has I q (K) CI 2 (K), where C > 0 is a absolue cosa. 6

7 If K is isoropic, oe ca chec ha q (K) c, where c > 0 is a absolue cosa (for a proof, see [25]). Therefore, (2.15) I q (K) C L K for every q. I paricular, from (2.12) ad (2.15) we see ha, for all q, (2.16) w(z q (K)) w q (Z q (K)) ql K Le C be a symmeric covex body i R. For every δ 1 we defie (2.17) d (C, δ) = max{q 1 : w(c) δw q (C)}. I was proved i [13] ad [15] ha (2.18) (C) cd (C, 2) 2.7. For every -dimesioal subspace F of R we deoe by E he orhogoal subspace of F. For every φ F \{0} we defie E + (φ) = {x spa{e, φ} : x, φ 0}. K. Ball (see [1] ad [19]) proved ha, if K is a ceered covex body of volume 1 i R he, for every q 0, he fucio ( ) 1 (2.19) φ φ 1+ q q+1 q+1 2 x, φ q dx K E + (φ) is he gauge fucio of a covex body B q (K, F ) o F. A basic ideiy from [25] saes ha for every F G, ad every q 1 we have ha ( ) 1/q + q (2.20) P F (Z q (K)) = B +q 1(K, F ) 1/+1/q Z q (B +q 1(K, F )). 2 I is a simple cosequece of Fubii s heorem ha if K is isoropic he B +1 (K, F ) is almos isoropic. Moreover, usig (2.20) oe ca chec ha Z q (B +1 (K, F )) (2.21) c 1 P F (Z q (K)) + q Z q (B +1 (K, F )) c 2 + q L B+1 (K,F ) L K L B+1 (K,F ) for all 1, q. I paricular, for all q we have (2.22) Z q (B +1 (K, F )) L B+1 (K,F ) P F (Z q (K)) L K Recall ha if A ad B are covex bodies i R, he he coverig umber N(A, B) of A by B is he smalles umber of raslaes of B whose uio covers A. A simple ad useful observaio is ha, if A ad B are boh symmeric ad if S (A, B) is he maximal umber of pois z i A which saisfy z i z j B for all i j, he (2.23) N(A, B) S (A, B) N(A, (/2)B). 7

8 3 Coverig umbers of projecios of L q -ceroid bodies Le K be a isoropic covex body i R. We firs give a aleraive proof of some esimaes o he coverig umbers N(Z q (K), ql K B 2 ) ha were recely obaied i [7]; hey improve upo previous esimaes from [6]. Proposiio 3.1. Le K be a isoropic covex body i R, le 1 q ad 1. The, (3.1) log N (Z q (K), c 1 ql K B2 q ) c c 3, where c 1, c 2, c 3 > 0 are absolue cosas. Noe ha he upper boud i (3.1) is of he order / 2 if /q ad of he order q/ if /q. Our sarig poi is a small ball probabiliy ype esimae which appears i [22, Fac 3.2(c)]: Lemma 3.2. Le θ S 1, 1 1 ad r e. The, ({ (3.2) µ, F G, : P F (θ) 2 1 }) ( ) e. r r Uder he resricio log N(C, B 2 ), Lemma 3.2 allows us o compare he coverig umbers N(C, B 2 ) of a covex body C wih he coverig umbers of is radom -dimesioal projecios. Lemma 3.3. Le C be a covex body i R, le r e, s > 0 ad 1 1. If N s := N(C, sb2 ), he here exiss F G, such ha µ, (F) 1 Ns 2 e /2 r ad (3.3) N for all F F. ( P F (C), s 2r ) B F N s Proof. Le N s = N(C, sb2 ). From (2.23) we see ha here exis z 1,..., z Ns C such ha z i z j 2 s for all 1 i, j N s, i j. Cosider he se {w m : 1 m Ns(Ns 1) 2 } of all differeces z i z j (i j). Noe ha w m 2 s for all m. Lemma 3.2 shows ha ({ (3.4) µ, F G, : P F (w m ) 2 1 }) r w m 2 ( ) e, r ad hece, (3.5) µ, ({ F : P F (w m ) 2 1 r w m 2 for all m }) 1 N 2 s e /2 r. 8

9 Le F be he subse of G, described i (3.5). The, for every F F ad all i j, (3.6) P F (z i ) P F (z j ) 2 1 r z i z j 2 s r. Sice P F (z i ) P F (C), he righ had side iequaliy of (2.23) implies ha ( (3.7) N P F (C), s ) 2r B F N s, as claimed. Fially, we will use he followig regulariy esimae for he coverig umbers of L q -ceroid bodies (see [6, Proposiio 3.1] for a proof of he firs iequaliy ad [9] for a proof of he secod oe): For all > 0 ad 1 q, (3.8) log N (Z q (K), c q ql K B2 ) + ad log N ( K K, L K B2 ), where c > 0 is a absolue cosa. Noe ha he upper boud i (3.8) is of he order / if /q ad of he order q/ if /q. Proof of Proposiio 3.1. We se s = c ql K ad N s := N(Z q (K), sb 2 ). Because of (3.8) we may assume ha 3 N s e c, ad he, we choose 1 so ha log N s 2 log N s. We disiguish wo cases: (a) Assume ha 1 /q. Applyig Lemma 3.3 wih r = e 3 we have ha, wih probabiliy greaer ha 1 Ns 2 e 5/2 1 e /2, a radom subspace F G, saisfies ( ) (3.9) 2 log N s log N P F (Z q (K)), c 1 s B F, where c 1 > 0 is a absolue cosa. If log N s q he we rivially ge log N s / 2 because q / 2. So, we may assume ha log N s q; i paricular, q. The, usig (2.21) we ge ( (3.10) log N Z q (B +1 (K, F )), c L ) B +1 (K,F ) 2 L K sb F. Observe ha s / qlk = c / c c/q. Therefore, applyig he esimae (3.8) for he -dimesioal isoropic covex body B +1 (K, F ), we ge (3.11) 2 c 2 / = c 2, 9

10 which shows ha (3.12) log N(Z q (K), ql K B 2 ) = log N s c 3 2, where c 3 = 4c 2 2. (b) Assume ha /q. We se p := q q. The, usig (2.8), we have ha N (Z q (K), ( q ql K B2 ) N p Z p(k), c 4 ) ql K B2 ( ) p N Z p (K), c 4 plk B2 q ( ) = N Z p (K), c 4 plk B2. p Applyig he resul of case (a) for Z p (K) wih = /p, we see ha N (Z q (K), ( ) ql K B2 ) N Z p (K), c 4 plk B2 p ( ) q e c5p = exp c 5, ad he proof is complee. Usig Proposiio 3.1 we ca obai aalogous upper bouds for he coverig umbers of P F (Z q (K)), where F G,. Proposiio 3.4. Le K be a isoropic covex body i R. For every 1 q <, for every F G, ad every 1, we have (3.13) log N (P F (Z q (K)), ql K B F ) c c 2 q, where c 1, c 2 > 0 are absolue cosas. Also, for every q, F G, ad 1, (3.14) log N (P F (Z q (K)), ql K B F ) c 3 q, where c 3 > 0 is a absolue cosa. Proof. (i) Le 1 q, F G, ad 1. From (2.22) we see ha (3.15) log N (P F (Z q (K)), ( ql K B F ) log N Z q (B +1 (K, F )), c ) ql B+1 (K,F ) B F, where c > 0 is a absolue cosa. Sice B +1 (K, F ) is almos isoropic, we may apply Proposiio 3.1 for B +1 (K, F ) i F : we have ( (3.16) log N Z q (B +1 (K, F )), c ) ql B+1 (K,F ) B F c c 2 q, 10

11 ad hece, (3.17) log N (P F (Z q (K)), ql K B F ) c c 2 q. (ii) Assume ha q ad F G,. The, usig (2.21) ad he fac ha Z q (C) cov{c, C}, for every 1 we wrie log N (P F (Z q (K)), ( cq ql K B F ) log N D +1(K, F ), ) ql B+1 (K,F ) B F log N ( q c 3, D +1 (K, F ), LB+1(K,F ) q B F where D +1 (K, F ) = B +1 (K, F ) B +1 (K, F ), usig i he ed he secod esimae of (3.8) for he isoropic covex body B +1 (K, F ). This complees he proof. Usig hese bouds we ca prove he exisece of direcios wih relaively small ψ 2 -orm o ay subspace of R. The depedece is beer as he dimesio icreases. ) Theorem 3.5. Le K be a isoropic covex body i R. (i) For every log 2 / log ad every F G, here exiss θ S F ha such (3.18), θ ψ2 C / L K, (ii) For every / log ad every F G, here exiss θ S F such ha (3.19), θ ψ2 C log L K, where C > 0 is a absolue cosa. Proof. For every ieger q 1 we defie he ormalized L q ceroid body K q of K by (3.20) K q = 1 qlk Z q (K), ad we cosider he covex body (3.21) T = cov log 2 K 2 i i=1. 11

12 The, for every F G, we have (3.22) P F (T ) = cov log 2 i=1 P F (K 2 i). We will use he followig sadard fac (see [6] for a proof): If A 1,..., A s are subses of RB 2, he for every > 0 we have (3.23) N(cov(A 1 A s ), 2B 2 ) ( ) s s cr N(A i, B2 ). i=1 We apply his o he ses A i = P F (K 2 i). Observe ha K 2 i c 1 2 i/2 B2, ad hece, N(A i, B F ) = 1 if c 1 2 i/2. Also, A i c 2 BF for all i. Usig Proposiio 3.4, for every 1 we ca wrie log 2 N(P F (T ), 2B F ) (c 2 ) log 2 N(P F (K 2 i), B F ) e c3 log2 exp C i=1 log 2 i=1 2 i/2 + C i ( ) e c3 log2 exp C + C 2 log(/2 ), where he secod erm appears oly if c 2. Now, we disiguish wo cases: (i) If log 2 / log we choose 0 = /. Observe ha 0 ( ) ( ) 2 2 (3.24) 2 0 ( ) log 2 = 2 0 log log log. = ad This implies ha N(P F (T ), /B F ) e c. I follows ha (3.25) P F (T ) C / B F. Therefore, here exiss θ S F such ha (3.26) h T (θ) = h PF (T )(θ) C /, which implies (3.27), θ 2 i C 2 i/2 / L K for every i = 1, 2,..., log 2. This easily implies (3.18). 12

13 (ii) If / log we choose 0 = log log. Observe ha log ad ( ) (3.28) 2 log 0 2 = ( ) 0 log log ( ) log log log. log This implies ha N(P F (T ), log B F ) e c ad, as i case (i), we see ha (3.29), θ 2 i C 2 i/2 log L K for every i = 1, 2,..., log 2. The resul follows. 0 = We close his Secio wih a sech of he proof of a aalogue of he esimae of Proposiio 3.1 for N(Z q (K), ql K B 2 ) for (0, 1). Proposiio 3.6. Le K be a isoropic covex body i R. If 1 q ad (0, 1), he (3.30) N (Z q (K), c 1 ( ql K B2 c2 ) ) ad (3.31) N (Z q (K), c 3 ( qb2 c4 ) ), where c i > 0 are absolue cosas. Proof. The lower boud is a cosequece of he esimae Z q (K) 1/ c q B2 1/ (see [17]). The, we wrie (3.32) N (Z q (K), c 1 qb2 ) Z ( q(k) c 1 qb2 c2 ). For he upper boud, we will use he fac (see [7, Secio 3] for he idea of his cosrucio) ha here exiss a isoropic covex body K 1 i R wih he followig properies: (i) N ( Z q (K), ql K B 2 ) N ( Zq (K 1 ), c 1 qb 2 ) for every > 0. (ii) c 2 qb 2 Z q (K 1 ) for all 1 q. (iii) Z q (K 1 ) 1/ c 3 q/ for all 1 q. Therefore, for every (0, 1) we have ( N Z q (K), 2 ) qlk B2 Z q(k 1 ) + qb 2 qb 2 cz q(k 1 ) qb2 ( c ), ad (3.30) is proved. 13

14 4 O he disribuio of he ψ 2 -orm From Theorem 3.5 we ca deduce a measure esimae for he se of direcios which saisfy a give ψ 2 -boud. We sar wih a simple lemma. Lemma 4.1. Le 1 ad le A be a subse of S 1 which saisfies A F for every F G,. The, for every ε > 0 we have (4.1) σ(a ε ) 1 2 where ( ε 2 ) 1, (4.2) A ε = { y S 1 : if{ y θ 2 : θ A} ε }. Proof. We wrie (4.3) σ(a ε ) = χ Aε (y) dσ(y) = S 1 G, S F χ Aε (y) dσ F (y) dµ, (F ), ad observe ha, sice A S F, he se A ε S F coais a cap C F (ε) = {y S F : y θ 0 2 ε} of Euclidea radius ε i S F. I follows ha (4.4) χ Aε (y) dσ F (y) σ F (C F (ε)) 1 ( ε ) 1, S F 2 2 by a well-ow esimae o he area of spherical caps, ad he resul follows. Remar. As he proof of he Lemma shows, he srog assumpio ha A F for every F G, is o really eeded for he esimae o σ(a ε ). Oe ca have pracically he same lower boud for σ(a ε ) uder he weaer assumpio ha A F for every F i a subse F, of G, wih measure µ, (F, ) c. Theorem 4.2. Le K be a isoropic covex body i R. For every log 2 here exiss A S 1 such ha c1 log (4.5) σ(a ) e where c 1 > 0 is a absolue cosa, ad { } (4.6), y ψ2 C max /, log for all y A. Proof. We fix log 2 / log ad defie A o be he se of θ S 1 which saisfy (3.18). By Theorem 3.5 we have A S F for every F G,. Therefore, we ca apply Lemma 4.1 wih ε = 1. If y A ε he here exiss θ A such ha y θ 2 ε, which implies (4.7), y θ ψ2 (, y θ, y θ ψ1 ) 1/2 c ε L K, L K 14

15 if we ae io accou he well-ow fac ha, θ ψ1 c, θ 1 cl K (see [19]) ad he fac ha, θ ( + 1)L K. I follows ha, y ψ2, θ ψ2 +, y θ ψ2, θ ψ2 + c / L K. Sice θ saisfies (3.18), we ge (4.6) wih a differe absolue cosa C for all y A := A 1/. Fially, Lemma 4.1 shows ha ( 1 ) 1 e c1 log, (4.8) σ(a ) which complees he proof i his case. A similar argume wors for / log : i his case, we apply Lemma 4.1 wih ε = log / ad he measure esimae for A is he same. Proof of Theorem 1.1: Le 1 ad cosider he larges for which / log. The, (4.9) log log, 2 ad hece, e c1 log e c2/2. Theorem 4.2 shows ha (4.10) ψ K () σ(a ) e c2/2. This proves our claim. 5 O he mea widh of L q ceroid bodies 5.1. Mea widh of Z q (K). Le K be a isoropic covex body i R. For every q q (K) we have (5.1) w(z q (K)) w q (Z q (K)) q/i q (K) c ql K. Sice q (K) c, (5.1) holds a leas for all q. For q, we may use he fac ha Z q (K) c(q/ )Z (K) o wrie (5.2) w(z q (K)) c q w(z (K)) c 1 q 4 L K. I oher words, for all q 1 we have (5.3) w(z q (K)) c ql K (1 + ) q 4. Seig q = ad aig io accou (2.8) we ge he geeral upper boud (5.4) w(k) c 1 w(z (K)) c 2 3/4 L K 15

16 for he mea widh of K. I he ex Proposiio we slighly improve hese esimaes, aig io accou he radius of Z q (K) or K. Proposiio 5.1. Le K be a isoropic covex body i R ad le 1 q /2. The, (5.5) w(z q (K)) c ql K (1 + R(Z q (K))/ ) L K. I paricular, (5.6) w(k) c L K (1 + R(K)/ ) L K. Proof. Recall ha, for all 1 q /2, (5.7) I q (K) /qw q (Z q (K)). We firs observe ha, for every 1, (5.8) w q/ 2(Z q (K)) c 2 w q/ 2(Z q/ 2(K)) 2 q 2 I q/ 2(K) c ql K. Le δ 1. Recall ha d (C, δ) = max{q 1 : w(c) δw q (C)}. We disiguish wo cases: (a) If q d (Z q (K), δ) he, by (5.7), we have ha (5.9) w(z q (K)) δw q (Z q (K)) δ qi q (K)/ cδ ql K. (b) If q d (Z q (K), δ), we se d := d (Z q (K), δ) ad defie 1 by he equaio q/ 2 = d. The, usig (5.8), we have (5.10) w(z q (K)) δw d (Z q (K)) = δw q/ 2(Z q (K)) cδ ql K. This gives he boud q (5.11) w(z q (K)) cδ d (Z q (K), δ) L K. Moreover, usig he fac ha (5.12) d (Z q (K), c 2 ) (Z q (K)) w(z q(k))2 R(Z q (K)) 2, we see ha if if q c 1 d (Z q (K), c 2 ) he q R(Zq (K)) (5.13) w(z q (K)) c 4 LK. 16

17 Choosig δ = 2 ad combiig he esimaes (5.9) ad (5.13) we ge (5.5). Seig q = ad usig (2.8) we obai (5.6). Recall ha K is called a ψ α -body wih cosa b α if (5.14), θ ψα b α, θ 1 for all θ S 1. If we assume ha K is a ψ α body for some α [1, 2] he R(Z q (K)) R(b α q 1/α Z 2 (K)) = b α q 1/α L K, ad Proposiio 5.1 gives immediaely he followig. Proposiio 5.2. Le K be a isoropic covex body i R. If K is a ψ α -body wih cosa b α for some α [1, 2] he, for all 1 q, (5.15) w(z q (K)) c ) bα q ql K (1 1 2α + 4 ad (5.16) w(k) c b α α+2 4α LK Mea widh of Ψ 2 (K). As a applicaio of Theorem 1.1 we ca give he followig esimae for he q-widh of Ψ 2 (K) for egaive values of q. Proposiio 5.3. Le K be a isoropic covex body i R ad 1. The (5.17) w 2 (Ψ 2(K)) c log L K. Proof. Observe ha, by Marov s iequaliy, ( (5.18) σ {θ S 1 : h Ψ2(K)(θ) 1 ) e w (Ψ 2(K))} e 2 2. From Theorem 1.1 we ow ha ( (5.19) e 2 σ {θ S 1 : h Ψ2(K)(θ) c ) log L K }, for some absolue cosa c > 0. This proves (5.17). We ca also give a upper boud for he mea widh of Ψ 2 (K): Proposiio 5.4. Le K be a isoropic covex body i R. The, (5.20) w(ψ 2 (K)) c 4 log L K. Proof. Le w := w(ψ 2 (K)). Sice R(Ψ 2 (K)) c L K, usig (2.18) we see ha (5.21) d (Ψ 2 (K)) c (Ψ 2 (K)) c w2 L 2. K 17

18 We choose so ha = c w2, i.e. 2 L 2 K (5.22) = c L K w The, from Proposiio 5.3 we see ha 1. w cw d (Ψ 2 (K)) w cw 2 ad (5.20) follows. c 1 log L 2 w K, L 2 K (Ψ 2 (K)) = w 2 (Ψ 2(K)) Acually, we ca remove he logarihmic erm, sarig wih he ex lemma: Lemma 5.5. Le K be a isoropic covex body i R ad le 1 1. The for every F G,, (5.23) P F (Ψ 2 (K)) c / where c > 0 is a absolue cosa. L K L B+1 (K,F ) Ψ 2 (B +1 (K, F )), Proof. Ideed, because of (2.21) ad (2.22), for every θ S F we ca wrie h Ψ2(K)(θ) h Zq(K)(θ) h Zq(K)(θ) sup + sup L K 1 q qlk q qlk h PF (Z = sup q(k))(θ) h PF (Z + sup q(k))(θ) 1 q qlk q qlk h Zq(B c 1 sup +1 (K,F )) (θ) + c 2 sup 1 q qlb+1(k,f ) q h Zq(B = c 1 sup +1 (K,F )) (θ) + c 2 sup 1 q qlb+1(k,f ) q c 3 h Ψ2(B +1 (K,F )) (θ) L B+1 (K,F ) h Ψ2(B c +1 (K,F )) (θ) 5. L B+1 (K,F ) h PF (Z (K))(θ) qlk q q h Z (B +1 (K,F )) (θ) LB+1(K,F ) q h Ψ2(B + c 4 sup +1 (K,F )) (θ) q L B+1 (K,F ) Proposiio 5.6. Le K be a isoropic covex body i R. The (5.24) w(ψ 2 (K)) c 4 L K. 18

19 Proof. Le =. Usig Lemma 5.5 we see ha w(ψ 2 (K)) = w(p F (Ψ 2 (K)))dµ, (F ) G, L K c w(ψ 2 (B +1 (K, F )))dµ, (F ). G, L B+1 (K,F ) Sice = q (K), we ow ha a radom B +1 (K, F ) is ψ 2 (see [8]), ad he resul follows. Applyig Lemma 5.5 we ca cover he case 1 log 2 i Theorem 3.5: Corollary 5.7. Le K be a isoropic covex body i R. For every 1 log 2 ad every F G, here exiss θ S F such ha (5.25), θ ψ2 C / log 2 L K, where C > 0 is a absolue cosa. I fac, for a radom F G, he erm log 2 is o eeded i (5.25). Proof. Le 1 log 2 ad F G,. Sice B +1 (K, F ) is isoropic, Theorem 3.5(ii) shows ha here exiss θ S F such ha (5.26) h Ψ2(B +1 (K,F )) (θ) c 1 log 2LB+1 (K,F ). The, Lemma 5.5 shows ha, θ ψ2 h Ψ2(K)(θ) = h PF (Ψ 2(K))(θ) c / L K L B+1 (K,F ) h Ψ2(B +1 (K,F )) (θ) C / log 2 L K. I fac, sice log 2 q (K), for a radom F G, we ow ha B +1 (K, F )) is a ψ 2 -body (see [8]), ad hece, h Ψ2(B +1 (K,F )) (θ) c 2L B+1 (K,F ) for all θ S F. Usig his esimae isead of (5.26) we may remove he log 2-erm i (5.25) for a radom F G,. Proof of Proposiio 1.3. Sice h Ψ2(K) is L K -Lipschiz, we have ha (5.27) σ ( {θ S 1 : h Ψ2(K)(θ) w(ψ 2 (K)) sw(ψ 2 (K))} ) e cs2( w(ψ 2 (K)) LK ) 2. Le u 2w(Ψ 2 (K)). The, u = (1 + s)w(ψ 2 (K)) for some s 1 ad sw(ψ 2 (K)) u/2. From (5.27) i follows ha (5.28) σ ( {θ S 1 : h Ψ2(K)(θ) u} ) exp ( cu 2 /L 2 K). If c 4 1 / log, he Proposiio 5.7 shows ha u = log L K 2w(Ψ 2 (K)). The, we ca apply (5.28) o ge he resul. The esimae of Proposiio 1.3 holds rue for all cw(ψ 2 (K))/ log L K ; his is easily checed from he proof. This shows ha beer lower bouds for ψ K () would follow from a beer upper esimae for w(ψ 2 (K)) ad vice versa. 19

20 6 O he mea widh of isoropic covex bodies Le K be a isoropic covex body i R. For every 2 q we defie ( ) 2 w(zq (K)) (6.1) (q) =. R(Z q (K)) Sice, θ q cql K for all θ S 1, we have R(Z q (K)) cql K. Therefore, (6.2) w(z q (K)) cql K (q) The, from (2.8) we see ha (6.3) w(k) w(z (K)) c q w(z q(k)) c (q) L K. Defie (6.4) ρ = ρ (K) = mi 2 q (q). Sice q was arbirary i (6.3), we ge he followig: Proposiio 6.1. For every isoropic covex body K i R oe has (6.5) w(k) c ρ (K) L K. Our ex observaio is he followig: by he isoperimeric iequaliy o S 1, for every q 1 oe has ( (6.6) σ, θ q w(z q ) w(z ) q) exp( c (q)) exp( 2cρ ) 2 where c > 0 is a absolue cosa. Assume ha log e cρ. The, (6.7), θ w(z q ) for all θ o a subse A q of S 1 of measure σ(a q ) 1 exp( cρ ). Taig q i = 2 i, i log 2 ad seig A = A qi, we have he followig: Lemma 6.2. For every isoropic covex body K i R wih ρ (K) C log log oe ca fid A S 1 wih σ(a) 1 e cρ such ha (6.8), θ q w(z q ) for all θ A ad all 2 q. I paricular, w(z q ) (6.9), θ ψ2 max 2 q q for all θ A. 20

21 Lemma 6.2 implies ha if ρ (K) is large ad, θ ψ2 is well-bouded o a relaively large subse of he sphere, he a similar boud holds rue for almos all direcios. As a cosequece, we ge a good boud for he mea widh of K. The precise saeme is he followig. Proposiio 6.3. Le K be a isoropic covex body i R which saisfies he followig wo codiios: The, 1. ρ (K) C log log. 2. For some b > 0 we have, θ ψ2 b L K for all θ i a se B S 1 wih σ(b) > e cρ. (6.10), θ ψ2 Cb L K for all θ i a se A S 1 wih σ(a) > 1 e cρ. Also, (6.11) w(z q (K)) c qb L K for all 2 q ad (6.12) w(k) C b L K. Proof. We ca fid u A B, where A is he se i Lemma 6.2. Sice u B, we have (6.13), u q C 1 qb L K for all 2 q, ad (6.8) shows ha (6.14) w(z q (K)) C 2 qb L K for all 2 q. Goig bac o (6.8) we see ha if θ A he (6.15), θ q cw(z q ) C 3 qb L K for all 2 q. For q = we ge (6.12). Fially, for every θ A we have, θ q (6.16), θ ψ2 max Cb L K. 2 q q This complees he proof. Proposiios 6.1 ad 6.3 provide a dichoomy. If ρ (K) is small he we ca use Proposiio 6.1 o ge a upper boud for w(k). If ρ (K) is large he we ca use Proposiio 6.3 provided ha we have some sufficiely good lower boud for ψ K (): wha we have is (6.17) ψ K () e c1/2 e cρ, 21

22 if /ρ. Therefore, we obai he esimae (6.18) w(k) C log /ρ L K. Combiig he previous resuls, we deduce oe more geeral upper boud for he mea widh of K. Theorem 6.4. For every isoropic covex body K i R we have (6.19) w(k) C { ρ mi, } log /ρ L K, where c > 0 is a absolue cosa. The esimae i Theorem 6.4 depeds o our owledge for he behavior of ψ K (); as i sads, i oly recovers he O( 3/4 L K ) boud for he mea widh of K. Acually, he logarihmic erm i (6.19) maes i slighly worse. However, we ca remove his logarihmic erm, sarig wih he followig modificaio of Proposiio 5.1. Proposiio 6.5. Le K be a isoropic covex body i R ad 1 q. The, (6.20) w(z q (K)) c ) q ql K (1 +, (Z q (K)) where c > 0 is a absolue cosa. Proof. If R(Z q (K)) c L K he (6.20) is a direc cosequece of (5.5). So, we assume ha R(Z q (K)) c L K. The, wriig (5.5) i he form q (6.21) w(z q (K)) c 4 R(Z q (K)) L K, ad aig io accou he defiiio of (Z q (K)) we see ha (6.22) R(Z q (K)) LK c 1 q (Z q (K)), ad (6.20) follows from (5.5) agai. Theorem 6.6. Le K be a isoropic covex body i R. The, (6.23) w(k) c { } ρ L K mi,, where c > 0 is a absolue cosa. Proof. From Proposiio 6.1 we ow ha (6.24) w(k) c L K ρ. ρ 22

23 Le q 0 saisfy ρ = (Z q0 (K)). From Proposiio 6.5 ad from (2.7) ad (2.8) we have ha, for all 1 q, (6.25) w(k) c ( ) q w(z q(k)) c 1 LK q +. (Z q (K)) Recall ha q is he parameer q (K) := max{q [1, ] : (Z q (K)) q}. We disiguish wo cases. (i) Assume ha q 0 q. The we apply (6.25) for q ; sice q = (Z q (K)) ρ, we ge (6.26) w(k) 2c 1 LK q 2c 1 LK ρ. (ii) Assume ha q 0 q. The, q 0 (Z q0 (K)) = ρ. Applyig (6.25) for q 0, we ge (6.27) w(k) 2c 1 LK (Z q0 (K)) = 2c 1 LK. ρ I boh cases, we have (6.28) w(k) c L K ρ. Combiig (6.28) wih (6.24) we ge he resul. Acowledgeme. We would lie o ha he referee for useful commes regardig he preseaio of his paper. Refereces [1] K. M. Ball, Logarihmically cocave fucios ad secios of covex ses i R, Sudia Mah. 88 (1988), [2] S. G. Bobov ad F. L. Nazarov, O covex bodies ad log-cocave probabiliy measures wih ucodiioal basis, Geom. Aspecs of Fuc. Aalysis (Milma- Schechma eds.), Lecure Noes i Mah (2003), [3] S. G. Bobov ad F. L. Nazarov, Large deviaios of ypical liear fucioals o a covex body wih ucodiioal basis, Sochasic Iequaliies ad Applicaios, Progr. Probab. 56, Birhauser, Basel (2003), [4] J. Bourgai, O he disribuio of polyomials o high dimesioal covex ses, Lecure Noes i Mahemaics 1469, Spriger, Berli (1991), [5] A. Giaopoulos ad V. D. Milma, Mea widh ad diameer of proporioal secios of a symmeric covex body, J. Reie Agew. Mah. 497 (1998),

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25 [27] G. Pisier, The Volume of Covex Bodies ad Baach Space Geomery, Cambridge Tracs i Mahemaics 94 (1989). [28] P. Pivovarov, O he volume of caps ad boudig he mea-widh of a isoropic covex body, Mah. Proc. Cambridge Philos. Soc. 149 (2010), A. Giaopoulos: Deparme of Mahemaics, Uiversiy of Ahes, Paepisimioupolis , Ahes, Greece. apgiaop@mah.uoa.gr G. Paouris: Deparme of Mahemaics, Texas A & M Uiversiy, College Saio, TX U.S.A. grigoris paouris@yahoo.co.u P. Valeas: Deparme of Mahemaics, Uiversiy of Ahes, Paepisimioupolis , Ahes, Greece. pevale@mah.uoa.gr 25