About the isotropy constant of random convex sets

ABOUT THE ISOTR OPY CONST ANT OF RANDOM CONVEX SETS DA VID ALONSO-GUTI ´ ERREZ* Abstract. Let K be the symmetric conv ex hu ll of m indep enden t random v ectors unif ormly distributed on the unit sphere of R n . W e prov e that, for ev ery δ > 0, the i sotrop y constan t of K is bounded b y a c onstan t c ( δ ) with high pr obabili t y , p rovided that m ≥ (1 + δ ) n . This result answers a question raised in [4]. 1. Introduction and not a tion A conv ex b o dy K ⊂ R n is a compact conv ex set with non-empty interior. A conv ex bo dy is s aid to be in isotropic p ositio n if it has volume 1 and sa tisfies the following tw o conditions: • R K xdx = 0 (cent er of mas s at 0) • R K h x, θ i 2 dx = L 2 K ∀ θ ∈ S n − 1 where L K is a constant indep endent of θ , which is calle d the isotropy co nstant of K . Here h· , ·i denotes the standard sca lar pro duct in R n . It is well kno wn that for ev ery conv ex bo dy K ∈ R n there exists an affine map T such that T K is isotropic. F urthermore, K a nd T K are b oth isotro pic if and o nly if T is an orthog o nal transfor mation. In this case the isotropy constant of bo th K and T K is the same. Hence, we can define the is otropy constant for every conv ex bo dy . It v erifies the following equation: nL 2 K = min  1 | T K | 1+ 2 n Z T K | x | 2 dx | T ∈ GL ( n )  , where | · | denotes b o th the volume of a se t in R n and the e uclidean norm of a v ector (see [5]). It is conjectured that ther e exists an absolute cons tant C such that fo r every conv ex b o dy K , L K ≤ C . This conjecture has b een verified for s everal classes of conv ex bo dies such as unco nditional b o dies, zonoids and others. Howev er, the b est known general upp er b ound is L K ≤ cn 1 4 ,[3], which improv es the ea rlier estimate L K ≤ c n 1 4 log n given by B ourgain (see [1]). In a recent paper [4], Kla r tag and Kozma proved that with high pr obability the co njecture is true for the convex h ull of indep endent gauss ia n v ectors and suggested that with the sa me techniques it should b e p ossible to obtain a n analo gous result for the conv ex h ull of indep endent po int s uniformly distributed o n the sphere (in this case the co ordina tes are not Date : July 2007. *Supported by FPI Scholarship f rom DGA (Spain). 1 2 DA VID ALONSO-GUTI ´ ERREZ* independent). In this pap er we are going to pr ov e this result. T o b e precise, we are going to prov e the following theorem: Theorem 1.1. F or every δ > 0 , ther e exist c onstants c ( δ ) , c 1 and c 2 , such that if m > (1 + δ ) n , { P i } m i =1 ar e indep e ndent r andom ve ctors on S n − 1 and K = c onv {± P 1 , . . . , ± P m } , then P { L K ≤ c ( δ ) } ≥ 1 − c 1 e − c 2 n min { 1 , log m n } . Along this pa p er we will denote by σ the uniform pr obability measure on the sphere S n − 1 and by ω n the volume of the euclidean ball. ∆ n will b e the regular simplex of dimension n and the letter s C , c, c 1 , c 2 , . . . will always denote a bsolute constants whose v alue may change from line to line. 2. Some previous resul ts In this s ection we are g oing to reca ll some known facts that we are going to use. W e b eg in with the next Definition 2. 1 . We write L ψ 2 for the sp ac e of r e al value d me asur able functions on S n − 1 such that R S n − 1 e | f | 2 λ 2 dσ < ∞ for some λ > 0 , and we set k f k ψ 2 = inf  λ > 0 : Z S n − 1 e | f | 2 λ 2 dσ < 2  . Bernstein’s inequality g ives a bo und for the probability that the absolute v alue of the sum o f N indep endent random v a riables with mean 0 is bigg er than ǫN for certain v alues of ǫ . There ar e several versions of this ine q uality depending on the space wher e these r andom v ariables b elong to . W e are going to us e the follo wing version, who s e pro o f can b e found in [2]. Theorem 2.1. (Bernstein ’s ine quality) Assume { g j } N j =1 ⊂ L ψ 2 such t hat k g j k L ψ 2 ≤ A for al l j and some c onstant A . Then, for al l ǫ > 0 , P          N X j =1 g j       > ǫN    ≤ 2 e − ǫ 2 N 8 A 2 . W e will apply this inequality to the indep endent identically dis tr ibuted rando m v ariables √ n h P, θ i where P is a p oint distributed uniformly in the unit sphere. T o see that these ra ndom v ariables ar e in L ψ 2 we need the following le mma : Lemma 2.1. F or al l q ≥ 1 and for al l θ ∈ S n − 1 Z S n − 1 |h u, θ i| q dσ ( u ) = 2Γ  1+ q 2  Γ  1 + n 2  √ π n Γ  n + q 2  . ABOUT THE ISOTROPY CONST ANT OF RANDOM CONVEX S E TS 3 Pr o of. Cha nging to po lar c o ordinates in the follo wing integral, we hav e that Z R n |h x, θ i| q e − | x | 2 2  √ 2 π  n dx = nω n  √ 2 π  n Z ∞ 0 r n + q − 1 e − r 2 2 dr Z S n − 1 |h u, θ i| q dσ ( u ) = 2 q 2 n Γ  n + q 2  2Γ  1 + n 2  Z S n − 1 |h u, θ i| q dσ ( u ) . On the other hand, since Z R n |h x, θ i| q e − | x | 2 2  √ 2 π  n dx = Z ∞ −∞ | x | q e − x 2 2 √ 2 π dx = 2 √ 2 π Z ∞ 0 x q e − x 2 2 dx = 2 q 2 Γ  1+ q 2  √ π , we obtain the re s ult.  R emark. By Stirling’s for mu la w e have that for all q ≥ 1 and for all θ ∈ S n − 1 :  Z S n − 1 |h u, θ i| q dσ ( u )  1 q ∼ r q q + n . Corollary 2. 1. Ther e exists a c o nstant A > 0 such that for every θ ∈ S n − 1 the functional √ n h P, θ i s atisfies k √ n h· , θ ik ψ 2 ≤ A . Pr o of. Le t θ b e a unit vector, then Z S n − 1 e n |h P,θ i| 2 A 2 dσ ( P ) = ∞ X q =0 Z S n − 1 n q |h P, θ i| 2 q A 2 q q ! ≤ 1 + ∞ X q =1 C 2 q (2 q ) q A 2 q q ! ≤ 1 + C ′ ∞ X q =1  2 C 2 e A 2  q 1 √ 2 π q ≤ 1 + C ′ 1 1 − 2 C 2 e A 2 − 1 ! ≤ 2 if A > 0 is c hosen large enough.  4 DA VID ALONSO-GUTI ´ ERREZ* 3. Symmetric convex hull of random points in the sphere In this section w e ar e go ing to prov e theo rem 1.1. As it was said in the introduction, the isotropy constant of every co nv ex b o dy verifies the following equation: nL 2 K = min  1 | T K | 1+ 2 n Z T K | x | 2 dx | T ∈ GL ( n )  so, in particular nL 2 K ≤ 1 | K | 2 n 1 | K | Z K | x | 2 dx W e will prove our result in tw o steps: we will give a lower b ound for | K | 1 n and an upper b ound for 1 | K | R K | x | 2 dx which hold with high probability and this will imply o ur statement. Lemma 3.1. F or every δ > 0 , ther e exists a c onstant c ( δ ) such that if (1 + δ ) n < m < ne n 2 , { P i } m i =1 ar e indep endent r andom ve ctors on S n − 1 , and K = c onv {± P 1 , . . . , ± P m } , then | K | 1 n ≥ c ( δ ) p log m n n with pr ob ability gr e ater than 1 − e − n . Pr o of. Fir st observe that, with pro bability 1, the facets o f K a re simplices. Let α ∈ (0 , 1). If αB n 2 * K then there exists a facet of K , which lies in a hyper plane orthogo nal to some v ector θ ∈ S n − 1 such that |h P i , θ i| ≤ α for all i . Let us denote this facet conv { Q 1 , . . . , Q n } w ith Q j ∈ {± P 1 , . . . , ± P m } and with Q i 6 = ± Q j . It follows that P { αB n 2 * K } ≤  2 m n  P { P ∈ S n − 1 : |h P, θ i| ≤ α } m − n If c √ n ≤ α ≤ 1 4 then P { P ∈ S n − 1 : |h P, θ i| > α } = 2 ( n − 1 ) ω n − 1 nω n Z 1 α (1 − x 2 ) n − 3 2 ≥ 2 ( n − 1 ) ω n − 1 nω n Z 2 α α (1 − x 2 ) n − 3 2 ≥ 2 ( n − 1 ) ω n − 1 nω n α (1 − 4 α 2 ) n − 3 2 ≥ c ′ (1 − 4 α 2 ) n − 3 2 = c ′ e n − 3 2 log(1 − 4 α 2 ) ≥ c ′ e − 4 α 2 n . ABOUT THE ISOTROPY CONST ANT OF RANDOM CONVEX S E TS 5 So w e have P { αB n 2 * K } ≤  2 m n   1 − c ′ e − 4 α 2 n  m − n ≤  2 em n  n e ( m − n ) log (1 − c ′ e − 4 α 2 n ) ≤  2 em n  n e − c ′ ( m − n ) e − 4 α 2 n Now, set x = m n . Using the fact that there exits a co ns tant C such that √ x < c ′ ( x − 1) log x +2+log 2 for all x > C ,one can c hec k tha t if C n ≤ m ≤ ne n 2 and α = 1 2 √ 2 q log m n n then P ( 1 2 √ 2 r log m n n B n 2 * K ) ≤ e − n . On the other hand, if (1 + δ 0 ) n ≤ m ≤ C n , since P { P ∈ S n − 1 : |h P, θ i| ≤ ǫ } = 2( n − 1) ω n − 1 nω n Z ǫ 0 (1 − x 2 ) n − 3 2 dx ≤ c √ nǫ, we hav e that if c 1 = c 1 ( δ ) is chosen small enough P ( c 1 r log m n n B n 2 * K ) ≤  2 em n  n  cc 1 r log m n  m − n ≤ (2 e C ) n ( cc 1 ) m − n (log C ) C n 2 ≤  2 eC ( cc 1 ) δ (log C C 2 )  n ≤ e − n . Hence P ( min  c 1 ( δ ) , 1 2 √ 2  r log m n n B n 2 * K ) ≤ e − n and this completes the pro of.  Now let us give an upp er b ound for 1 | K | R K | x | 2 dx . It is stated in the next theorem whose pr o of follows the idea s in [4]. Theorem 3.1 . Ther e exist absolute c onstants C and C 1 such that if { P i } m i =1 ar e indep e ndent r andom ve ct ors on S n − 1 , m > n , and K = c onv {± P 1 , . . . , ± P m } then P  1 | K | Z K | x | 2 dx ≤ C log m n n  ≥ 1 − 2 e − C 1 n log m n . Pr o of. B y Ber nstein’s inequalit y , if { P i } n i =1 are indep e ndent rando m vectors on the sphere a nd if θ is a fixed p oint o n the spher e, then, for all ǫ > 0, P (      n X i =1 h P i , θ i      > ǫn ) ≤ 2 e − ǫ 2 n 2 8 A 2 . 6 DA VID ALONSO-GUTI ´ ERREZ* Now let N b e a 1 2 -net on the sphere such that |N | ≤ 5 n . Then P (      n X i =1 h P i , θ i      > ǫn for some θ ∈ N ) ≤ 2 e − ǫ 2 n 2 8 A 2 + n log 5 , and hence, P (      n X i =1 h P i , θ i      ≤ ǫn for every θ ∈ N ) ≥ 1 − 2 e − ǫ 2 n 2 8 A 2 + n log 5 Every θ ∈ S n − 1 can be written in the form θ = P ∞ j =1 δ j θ j , with θ j ∈ N and 0 ≤ δ j ≤  1 2  j − 1 so, if for every θ ∈ N it is true that | P n i =1 h P i , θ i| ≤ ǫn , then for every θ ∈ S n − 1      h n X i =1 P i , θ i      =       h n X i =1 P i , ∞ X j =1 δ j θ j i       ≤ ∞ X j =1 δ j |h n X i =1 P i , θ j i| ≤ 2 ǫn. Hence P (      n X i =1 P i      < 2 ǫn ) = P ( max θ ∈ S n − 1 h n X i =1 P i , θ i < 2 ǫn ) ≥ 1 − 2 e − ǫ 2 n 2 8 A 2 + n log 5 . Now, since P i 6 = j h P i , P j i = | P n i =1 P i | 2 − P n i =1 | P i | 2 = | P n i =1 P i | 2 − n , this implies that if ǫ > ǫ 0 > p 32 A 2 log 5 , then P    X i 6 = j h P i , P j i ≤ ǫn    ≥ 1 − 2 e − C ǫn . Since every facet is with pro bability o ne F k = con v { Q k 1 , . . . Q k n } with Q k i ∈ {± P 1 , . . . , ± P m } and with Q k i 6 = ± Q k j and s inc e P and − P hav e the same distribution, if we put F 1 , . . . , F l a complete list o f the ( n − 1)-dimensional face ts of K , then P    max k =1 ...l X Q k i 6 = Q k j h Q k i , Q k j i > ǫn lo g m n    ≤  2 m n  2 e − C ǫn log m n ≤ 2 e − C ǫn log m n + n log ( 2 em n ) whenever ǫ > ǫ 0 , so c ho osing a cons ta nt ǫ big enough w e hav e tha t P    max k =1 ...l X Q k i 6 = Q k j h Q k i , Q k j i > C n log m n    ≤ 2 e − C 1 n log m n . F or each facet F k = con v { Q k 1 , . . . , Q k n } , let T be the fo llowing linear trans forma- tion: T =    Q k 1 (1) . . . Q k n (1) . . . . . . Q k 1 ( n ) . . . Q k n ( n )    . ABOUT THE ISOTROPY CONST ANT OF RANDOM CONVEX S E TS 7 Then F k = T (∆ n − 1 ) a nd 1 |F k | Z F k | x | 2 dx = 1 | ∆ n − 1 | Z ∆ n − 1 | T x | 2 dx. Since T x =    P n i =1 Q k i (1) x i . . . P n i =1 Q i ( n ) x i    then | T x | 2 = n X j =1 n X i =1 Q k i ( j ) x i ! 2 = n X j =1 n X i 1 ,i 2 =1 Q k i 1 ( j ) Q k i 2 ( j ) x i 1 x i 2 so 1 |F k | Z F k | x | 2 dx = 1 | ∆ n − 1 | n X j =1 n X i 1 ,i 2 =1 Q k i 1 ( j ) Q k i 2 ( j ) Z ∆ n − 1 x i 1 x i 2 dx. F rom the identit y 1 | ∆ n − 1 | Z ∆ n − 1 x i 1 x i 2 dx = 1 + δ i 1 i 2 n ( n + 1) this quantit y equals 1 n ( n + 1 ) n X j =1   n X i =1 2 Q k i ( j ) 2 + X i 1 6 = i 2 Q k i 1 ( j ) Q k i 2 ( j )   = 2 n + 1 + 1 n ( n + 1 ) X i 1 6 = i 2 h Q k i 1 , Q k i 2 i and ther e e x ist a bsolute constants C and C 1 such that the maxim um of this quantit y ov er all facets is less than C log m n n with probability bigger than 1 − 2 e − C 1 n log m n so sup i =1 ...l 1 |F i | Z F i | y | 2 dy ≤ C log m n n with pro bability bigger than 1 − 2 e − C 1 n log m n , where C and C 1 are a bsolute con- stants. But, in the same wa y as it is prov ed in [4] we have that • 1 |K| R K | x | 2 dx = 1 | K | P l i =1 d (0 , F i ) n +2 R F i | y | 2 dy • n | K | = P l i =1 d (0 , F i ) |F i | and hence 1 | K | Z K | x | 2 dx ≤ n n + 2 sup i =1 ...l 1 |F i | Z F i | y | 2 dy ≤ C log m n n with pr o bability grea ter than 1 − 2 e − C 1 n log m n .  Lemma 3 .1 a nd theorem 3 .1 imply that for every δ > 0 ther e exist a bsolute constants c ( δ ), c 1 , c 2 , C 1 , s uch that if (1 + δ ) n < m ≤ ne n 2 then P { L K ≤ c } ≥ 1 − 2 e − C 1 n log m n − e − n > 1 − c 1 e − c 2 n min { 1 , log m n } . Note that in ca se m > ne n 2 we hav e tha t P { 1 4 B n 2 * K } ≤ e − n 8 DA VID ALONSO-GUTI ´ ERREZ* so, with probability g r eater than 1 − e − n , nL 2 K ≤ 1 | K | 2 n 1 | K | Z K | x | 2 dx ≤ 1 | 1 4 B n 2 | 2 n ≤ cn and so the pr o of is complete. A CKNOWLEDGEMENTS This pap er was written while the author was in an ea rly stage r e searcher p o s i- tion of the r esearch training net work “Phenomena in High Dimensions” (MR TN- CT-2004 -5119 53) in Athens. The author would like to thank professor Apo stolos Giannop oulos for several helpful discussions as well as for his hospitality . References [1] Bourgain, J. On the distribution of p olynomials on high dimensional c onvex set s , Springer Lecture Notes in Math. 1469 (1991), pp. 127-137. [2] Bourgain, J.; Lindenstrauss, J.; Mi lman, V. D. Minkowski sums and symmetrizations . GAF A Seminar 86-87, Springer Lecture Notes in Math. 13 17 (19 88), pp.44-46 . [3] Klartag, B. On c onvex p erturb ati ons with a b ounde d isotr opic co nstant . Geom. F unct. Anal. 16 ,(2006), no.06, pp. 127 4-1290. [4] Klartag, B. ; Kozma, G. On the hyp erplan e c onje ctur e on r andom c onvex sets . (Preprint). [5] Mil man V.; Pa jor A. Isotr opic p os itions and inertia el lipsoids and zonoids of the unit b al l of a norme d n -dimensional sp ac e , GAF A Seminar 87-89, Springer Lecture Notes in Math. 1376 (1989) , pp. 64-104. E-mail addr ess : 498220@celes.uni zar.es Universidad de Zaragoza