ON THE CENTRAL LIMIT PROPERTY OF CONVEX BODIES

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1 ON THE CENTRAL LIMIT PROPERTY OF CONVEX BODIES S. G. Bobkov ad A. oldobsky February 1, 00 Abstract For isotropic covex bodies i R with isotropic costat L, we study the rate of covergece, as goes to ifiity, of the average volume of sectios of to the gaussia desity o the lie with variace L. Let be a isotropic covex body i R,, with volume oe. By the isotropy assumptio we mea that the bariceter of is at the origi, ad there exists a positive costat L so that, for every uit vector θ, x, θ dx = L. Itroduce the fuctio f (t) = vol 1 ( H θ (t)) dσ(θ), S 1 t R, expressig the average ( 1)-dimesioal volume of sectios of by hyperplaes H θ (t) = {x R : x, θ = t} perpedicular to θ S 1 at distace t from the origi (ad where σ is the ormalized uiform measure o the uit sphere). Whe the dimesio is large, the fuctio f is kow to be very close to the gaussia desity o the lie with mea zero ad variace L. Beig geeral ad iformal, this hypothesis eeds to be formalized ad verified, ad precise statemets may deped o certai additioal properties of covex bodies. For some special bodies, several types of closeess of f to gaussia desities were recetly studied i [B-V], cf. also [-L]. To treat the geeral case, the followig characteristic σ associated to turs out to be crucial: σ = Var( X ). L 4 Supported i part by the NSF grat DMS Supported i part by the NSF grat DMS

2 Here X is a radom vector uiformly distributed over, ad Var( X ) deotes the variace of X. I particular, we have the followig statemet which is proved i this ote. Theorem 1. For all 0 < t c, [ f 1 (t) e t /(L ) σ L C πl t + 1 ] (1) where c ad C are a positive umerical costats. Usig Bourgai s estimate L c log() 1/4 ([Bou], cf. also [D], [P]) the righthad side of (1) ca be bouded, up to a umerical costat, by σ log t 1/4 + 1, which is small for large up to the factor σ. Let us look at the behavior of this quatity i some caoical cases. For the -cube = [ 1, 1 ], by the idepedece of coordiates, σ = 4 5. For s the ormalized l 1 balls, σ = 1 ( + 1) ( + 3)( + 4) 1, as. Normalizatio coditio refers to vol () = 1, but a slightly more geeral defiitio σ = Var( X ) makes this quatity ivariat uder homotheties ad simplifies (E X ) computatios. For s the ormalized euclidea balls, σ = , as. Thus, σ ca be small ad moreover, i the space of ay fixed dimesio, the euclidea balls provide the miimum (cf. Theorem below). The property that σ is bouded by a absolute costat for all l p balls simultaeously was recetly observed by. Ball ad I. Perissiaki [B-P] who showed for these bodies that the covariaces cov(xi, Xj ) = EXi Xj EXi EXj are opositive. Sice i geeral Var( X ) = i=1 Var(Xi ) + i j cov(xi, Xj ), the above property together with the hichie-type iequality implies Var( X ) Var(Xi ) EXi 4 CL 4. i=1 i=1 The result was used i [A-B-P] to study the closeess of radom distributio fuctios F θ (t) = P{ X, θ t}, for most of θ o the sphere, to the ormal distributio

3 3 fuctio with variace L. This radomized versio of the cetral limit theorem origiates i the paper by V. N. Sudakov [S], cf. also [D-F], [W]. The reader may fid recet related results i [-L], [Bob], [N-R], [B-H-V-V]. It has become clear sice the work [S] that, i order to get closeess to ormality, the covexity assumptio does ot play a crucial role, ad oe rather eeds a dimesio free cocetratio of X aroud its mea. Clearly, the stregth of cocetratio ca be measured i terms of the variace of X, for example. Nevertheless, the questio o whether or ot the quatity σ ca be bouded by a uiversal costat i the geeral covex isotropic case is still ope, although it represets a rather weak form of aa-lovász-simoovits cojecture about Cheeger-type isoperimetric costats for covex bodies [-L-S]. For isotropic, the latter may equivaletly be expressed as the property that, for ay smooth fuctio g o R, for some absolute costat C, g(x) g(x) dx dx CL By a Cheeger s theorem, the above implies Poicaré-type iequality g(x) g(x) dx dx 4(CL ) g(x) dx. () g(x) dx which for g(x) = x becomes Var( X ) 16C L 4. That is, σ 16C. To boud a optimal C i (), R. aa, L. Lovász, ad M. Simoovits cosidered i particular the geometric characteristic χ() = χ (x) dx where χ (x) deotes the legth of the logest iterval lyig i with ceter at x. By applyig the localizatio lemma of [L-S], they proved that () holds true with CL = χ(). Therefore, σ L 8χ(), ad thus the right-had side of (1) ca also be bouded, up to a costat, by χ() t + 1. To prove Theorem 1, we eed the followig formula which also appears i [B-V, Lemma 1.]. Lemma 1. For all t, f (t) = Γ ( ) ( ) 3 1 ( ) π Γ 1 1 t dx. { x t } x x

4 4 For completeess, we prove it below (with a somewhat differet argumet). Proof. We may assume t 0. Deote by λ θ,t the Lebesgue measure o H θ (t). The λ t = λ θ,t dσ(θ) S 1 is a positive measure o R such that f (t) = λ t (). This measure has desity that is ivariat with respect to rotatios, i.e., dλ t dx = p t( x ), where p t is a fuctio o [t, ). To fid the fuctio p t, ote first that, for every r > t, r λ t (B(0, r)) = p t ( x ) dx = S 1 p t (s)s 1 ds, B(0,r) t where B(0, r) is the Euclidea ball with ceter at the origi ad radius r, ad S 1 = π/ is the surface area of the sphere Γ(/) S 1. O the other had, sice the sectio of B(0, r) by the hyperplae H θ (t) is the Euclidea ball i R 1 of radius (r t ) 1/, we have λ t (B(0, r)) = λ θ,t (B(0, r)) dσ(θ) = S 1 Takig the derivatives by r, we see that for every r t, which implies 1 π ( 1)/ Γ(1 + ( 1)/) (r t ) ( 1)/. (r t ) ( 1)/ r = π1/ Γ ( ) 1 Γ ( ) p t (r)r 1, p t (r) = Sice f (t) = λ t (), the result follows. Γ ( ) ( ) (r t ) ( 3)/ π Γ 1. r so Proof of Theorem 1. Let t > 0. By the Cauchy-Schwarz iequality, ( x L dx x L ) 1/= dx σ L, x L dx = x L x + L dx σ L. (3)

5 5 By Stirlig s formula, lim π Γ(/) πγ(( 1)/) = 1 Γ(/) so that the costats c = πγ(( 1)/)) appearig i Lemma 1 are O( ). Now, o the iterval [t, ) cosider the fuctio Its derivative g (z) = t ( 3) z 4 g (z) = 1 z ( ) ( 3)/ 1 t. z ( ) ( 5)/ 1 t 1 z z ( 1 t z ) ( 3)/ represets the differece of two o-egative terms. Both of them are equal to zero at t, ted to zero at ifiity ad each has oe critical poit, the first at z = t 1/, ad the secod at z = t 1/. Therefore, max z [t, ) g (z) 16 t ( 1). This implies that, for every x, x t, if L t, the ad by (3), where t = { x t}. Now, writig g ( x ) g ( L ) 16 x L t, ( 1) t g ( x ) g ( L ) dx 16σ L t ( 1), (4) f (t) = c t g ( x ) dx = c g ( L )vol ( t ) + c t (g ( x ) g ( L )) dx ad applyig (4), we see that, for all t L, f (t) c g ( L )vol ( t ) Cσ L t, where C is a umerical costat. This gives f (t) c g ( L ) c g ( L ) (1 vol ( t )) + Cσ L t. (5)

6 6 Recall that L c, for some uiversal c > 0 (the worst situatio is attaied at euclidea balls, cf. eg. [Ba]). Therefore (5) is fulfilled uder t c. To further boud the first term o the right had side of (5), ote that g (z) 1/z, so c g ( L ) C 0, for some umerical C 0. Also, if t c, ( ) 1 vol ( t ) vol (B(0, t)) = ω t c0 ( ) c <, where ω deotes the volume of the uit ball i R, ad where c 0 c ca be made less tha 1/ by choosig a proper c. This also shows that the first term i (5) will be domiated by the secod oe. Ideed, the iequality C 0 Cσ L t immediately follows from t c ad the lower boud o σ give i Theorem. Thus, f (t) c g ( L ) Cσ L t, ad we are left with task of comparig c g ( L ) with the gaussia desity o the lie. This is doe i the followig elemetary Lemma. If 0 t L, for some absolute C, Γ ( ) ( ) ( 3)/ ( ) πγ 1 1 t 1 1 L L πl e t /L C. Proof. Usig the fact that L is bouded from below, multiplyig the above iequality by πl ad replacig u = t /(L ), we are reduced to estimatig ( ) Γ ( ) ( ( ) Γ 1 1 u ) 3 e u Γ e u ( Γ 1 )e u ( ) Γ ( + ( ) Γ 1 e u 1 u ) 3. I order to estimate the first summad, use the asymptotic formula for the Γ- fuctio, Γ(x) = x x 1 e x πx ( x + O( 1 x ) ), as x +, to get Γ ( ) Γ ( ) = 1 ( ( ) ( 3)/ 1 e ( 1)/ ( ) = e 1/ 1 ) ( 3)/ e / π ( O( 1 ) ) 6 1( 1 + O π( 1) ( O( 1 ) ) 6( 1) ( )) 1.

7 7 Sice, by Taylor, ( ) 1 = e ( +1) log(1 1 ) = e ( 1/ 1 + O ( )) 1 1, the first summad is O( 1 ) uiformly over u 0. To estimate the secod summad, recall that 0 u /. ) 3 ψ (u) = e u ( 1 u u 0 [0, /] where ψ (u 0 ) = 0 (if it exists) satisfies ( 1 u ) 5 0 4). Hece, ψ (u 0 ) = u 0 3 proves Lemma. The fuctio satisfies ψ (0) = 0, ψ (/) = e /, ad the poit = 3 e u 0 (whe 3 e u 0 = O( 1 ), ad thus sup u ψ (u) = O( 1 ). This Remark. Returig to the iequality (1) of Theorem 1, it might be worthwhile to ote that, i the rage t c, the fuctio f satisfies, for some absolute C > 0, the estimate f (t) C /(CL t e t ) C c, ad i this sese it does ot eed to be compared with the Gaussia distributio i this rage. Ideed, it follows immediately from equality of Lemma 1 that f (t) C max z t g (z) P{ X t }, where X deotes a radom vector uiformly distributed over. Whe 3, i the iterval z t, the fuctio g (z) = 1 t (1 ) ( 3)/ attais its maximum at z z the poit z 0 = t where it takes the value g (z 0 ) 1 t. Hece, C max z t g (z) C t C c. O the other had, the probability P{ X t } ca be estimated with the help of Alesker s ψ -estimate, [A], Ee X /(C L ). We fiish this ote by a simple remark o the extremal property of the euclidea balls i the miimizatio problem for σ. Theorem. σ Proof. The distributio fuctio F (r) = vol ({x : x r}) of the radom vector X uiformly distributed i has desity F (r) = r 1 S 1 1 ( ) 1 r = S 1 r 1 σ r, r > 0. We oly use the property that q(r) = S 1 σ( 1 r ) is o-icreasig i r > 0. Clearly, this fuctio ca also be assumed to be absolutely cotiuous so that we

8 8 ca write + q(r) = r p(s) ds, r > 0, s for some o-egative measurable fuctio p o (0, + ). We have 1 = 0 df (r) = 0 r 1 1 p(s) q(r) dr = r 0<r<s s drds = p(s) ds. Hece, p represets a probability desity of a positive radom variable, say, ξ. Similarly, for every α >, Therefore, E X α = 0 r α+ 1 q(r) dr = s α p(s) ds = + α α Eξα. Var( X ) = = ( ) + 4 Eξ4 + Eξ 4 ( + 4)( + ) (Eξ ) Var(ξ ) 4 ( + 4)( + ) (Eξ ). Oe ca coclude that σ = Var( X ) (E X ) 4 (Eξ ) (+4)(+) ( ) = + Eξ Theorem follows. Ackowledgmet. We would like to thak V. D. Milma for stimulatig discussios. Refereces [A] S. Alesker. ψ -estimate for the Euclidea orm o a covex body i isotropic positio. Geom. Aspects Fuct. Aal. (Israel ), 1-4, Oper. Theory Adv. Appl., [A-B-P] M. Atilla,. Ball, ad I. Perissiaki. The cetral limit problem for covex bodies. Preprit (1998). [Ba]. Ball. Logaritmically cocave fuctios ad sectios of covex sets. Studia Math., 88 (1988),

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