A HYPERPLANE INEQUALITY FOR MEASURES OF CONVEX BODIES IN R n, n 4

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1 A HYPERPANE INEQUAITY FOR MEASURES OF CONVEX BODIES IN R, 4 AEXANDER ODOBSY Abstract. et 4. We show that for a arbitrary measure µ with eve cotiuous desity i R ad ay origi-symmetric covex body i R, µ() 1 B 1 B 1 max µ( ξ ) Vol () 1/, ξ S 1 where ξ is the cetral hyperplae i R perpedicular to ξ, ad B is the volume of the uit Euclidea ball i R. This iequality is sharp, ad it geeralizes the hyperplae iequality i dimesios up to four to the settig of arbitrary measures i place of volume. I order to prove this iequality, we first establish stability i the affirmative case of the Busema-Petty problem for arbitrary measures i the followig sese: if ε >, ad are origi-symmetric covex bodies i R, 4, ad the µ( ξ ) µ( ξ ) + ε, ξ S 1, µ() µ() + 1 B 1 B 1 Vol () 1/ ε. 1. Itroductio The hyperplae problem of Bourgai [Bo1], [Bo] asks whether there exists a absolute costat C so that for ay origi-symmetric covex body i R Vol () 1 C max ξ S 1 Vol 1( ξ ), (1) where ξ is the cetral hyperplae i R perpedicular to ξ. The problem is still ope, with the best-to-date estimate C 1/4 established by lartag [l], who slightly improved the previous estimate of Bourgai [Bo3]. We refer the reader to recet papers [E], [DP] for the history ad curret state of the hyperplae problem. 1

2 AEXANDER ODOBSY I the case where the dimesio 4, the iequality (1) ca be proved with the best possible costat (see [G3, Theorem ]): Vol () 1 B 1 B 1 max Vol 1( ξ ), () ξ S 1 with equality whe = B is the Euclidea ball. Here B = π / /Γ(1+/) is the volume of B. Throughout the paper, we deote the costat i () by c = B 1 B 1. Note that c < 1 for every N; this is a easy cosequece of the log-covexity of the Γ-fuctio. Iequality () follows from the affirmative aswer to the Busema- Petty problem i dimesios up to four. The Busema-Petty problem, posed i 1956 (see [BP]), asks the followig questio. Suppose that ad are origi-symmetric covex bodies i R such that for every ξ S 1, Vol 1 ( ξ ) Vol 1 ( ξ ). Does it follow that Vol () Vol ()? The aswer is affirmative if 4 ad egative if 5. The solutio was completed at the ed of the 9 s as the result of a sequece of papers [R], [Ba], [Gi], [Bo4], [], [Pa], [G1], [G], [Z1], [Z], [1], [], [Z3], [GS] ; see [3, p. 3] or [G3, p. 343] for the history of the solutio. Applyig the affirmative part of the solutio to the case where = B, oe immediately gets (). I this article we prove that iequality (1) holds i dimesios up to four with arbitrary measure i place of volume. et f be a eve cotiuous o-egative fuctio o R, ad deote by µ the measure o R with desity f. For every closed bouded set B R defie µ(b) = f(x) dx. Our extesio of () is as follows. B Theorem 1. If 4 ad is a origi-symmetric covex body i R, the µ() 1 c max µ( ξ ) Vol () 1/. (3) ξ S 1

3 HYPERPANE INEQUAITY FOR MEASURES 3 Moreover, the costat is the best possible, sice there exists a sequece of measures µ j with eve cotiuous desities such that lim j µ j (B ) max ξ S 1 µ j (B ξ ) Vol (B ) 1/ = 1 c. Zvavitch [Zv] foud a remarkable geeralizatio of the Busema- Petty problem to arbitrary measures, amely, oe ca replace volume by ay measure with eve cotiuous desity i R. I particular, if 4, the for ay origi-symmetric covex bodies ad i R the iequalities imply µ( ξ ) µ( ξ ), µ() µ(). ξ S 1 Zvavitch also proved that this is geerally ot true if 5, amely, for ay µ with strictly positive eve cotiuous desity there exist ad providig a couterexample. By aalogy with the volume case, oe would expect that this result immediately implies (3). The argumet, however, does ot work i this settig, because the measure µ of sectios of the Euclidea ball does ot have to be a costat. Istead, to prove (3) we establish stability i the affirmative part of Zvavitch s result i the followig sese, Theorem. et f be a eve o-egative cotiuous fuctio o R, 4, let µ be the measure with desity f, let ad be origi-symmetric covex bodies i R, ad let ε >. Suppose that for every ξ S 1, The µ( ξ ) µ( ξ ) + ε. (4) µ() µ() + Iterchagig ad, we get 1 c Vol () 1/ ε. (5) Corollary 1. Uder the coditios of Theorem, we have µ() µ() c 1 max µ( ξ ) µ( ξ ) { } max Vol () 1, Vol () 1. ξ S 1 (6)

4 4 AEXANDER ODOBSY Proof of Theorem 1. To prove the iequality (3) simply put = i Corollary 1. To show that the costat i (3) is sharp, let = B ad, for every j N, let f j be a o-egative cotiuous fuctio o [, 1] supported i (1 1, 1) ad such that 1 f j j(t)dt = 1. et µ j be the measure o R with desity f j ( ), where is the Euclidea orm. We have µ j (B ) = S 1 1 r 1 f j (r)dr, where S 1 = π / /Γ(/) is the surface area of the uit sphere i R. For every ξ S 1, µ j (B ξ ) = S 1 r f j (r)dr. Clearly, 1 lim r 1 f j (r)dr j 1 r f j (r)dr = 1. The result follows from the equality (use the formula Γ(x+1) = xγ(x)) S 1 S B = 1/ 1 c. It remais to prove Theorem. Note that stability i the origial Busema-Petty problem (for volume) was established i [5]. We discuss the relatio betwee differet stability estimates i the ed of the paper.. Prelimiaries We use the techiques of the Fourier approach to sectios of covex bodies; see [3] ad [Y] for details. As usual, we deote by S(R ) the Schwartz space of rapidly decreasig ifiitely differetiable fuctios (test fuctios) i R, ad S (R ) is the space of distributios over S(R ). Suppose that f is a locally itegrable complex-valued fuctio o R with power growth at ifiity, i.e. there exists a umber β > so that f(x) lim x β =. The f represets a distributio actig by itegratio: for every φ S, f,φ = f(x)φ(x) dx. R

5 HYPERPANE INEQUAITY FOR MEASURES 5 The Fourier trasform of a distributio f is defied by ˆf,φ = f, ˆφ for every test fuctio φ. A distributio f is called eve homogeeous of degree p R if f(x),φ(x/α) = α +p f,φ for every test fuctio φ ad every α R, α. The Fourier trasform of a eve homogeeous distributio of degree p is a eve homogeeous distributio of degree p. We say that a distributio is positive defiite if its Fourier trasform is a positive distributio i the sese that ˆf,φ for every o-egative test fuctio φ. Schwartz s geeralizatio of Bocher s theorem (see, for example, [GV, p.15]) states that a distributio is positive defiite if ad oly if it is the Fourier trasform of a tempered measure o R. Recall that a (o-egative, ot ecessarily fiite) measure µ is called tempered if R (1 + ) β dµ(x) < for some β >. For a origi-symmetric covex body i R we deote by x = mi{a : x a}, x R the orm i R geerated by. Our defiitio of a covex body assumes that the origi is a iterior poit of. If < p <, the p is a locally itegrable fuctio o R ad represets a eve homogeeous of degree p distributio. If p represets a positive defiite distributio for some p (,), the its Fourier trasform is a tempered measure which is at the same time a homogeeous distributio of degree +p. Oe ca express such a measure i polar coordiates, as follows. Propositio 1. ([3, Corollary.6]) et be a origi-symmetric covex body i R ad p (,). The fuctio p represets a positive defiite distributio o R if ad oly if there exists a fiite Borel measure µ o S 1 so that for every eve test fuctio φ, ( ) x p φ(x) dx = t p 1 ˆφ(tξ)dt dµ (ξ). R S 1 The followig propositio was proved i [GS] (see [3, Corollary 4.9]). Propositio. If 4 ad is ay origi-symmetric covex body i R, the the fuctio 1 represets a positive defiite distributio.

6 6 AEXANDER ODOBSY For ay eve cotiuous fuctio f o the sphere S 1 ad ay ozero umber p R, we deote by f r p the extesio of f to a eve homogeeous fuctio of degree p o R defied as follows. If x R, the x = rθ, where r = ad θ = x/. We put f r p (x) = f (θ)r p. It was proved i [3, emma 3.7] that the Fourier trasform of f r +1 is equal to aother cotiuous fuctio g o S 1 exteded to a eve homogeeous of degree 1 fuctio g r 1 o the whole of R (i fact, g is the spherical Rado trasform of f, up to a costat). This is why we ca remove smoothess coditios i the Parseval formula o the sphere [3, Corollary 3.3] ad formulate it as follows. Propositio 3. et be a origi-symmetric covex body i R. Suppose that 1 is a positive defiite distributio, ad let µ be the fiite Borel measure o S 1 that correspods to 1 by Propositio 1. The for ay eve cotiuous fuctio f o S 1, (f r +1 ) (θ) dµ (θ) = θ 1 f(θ) dθ. (7) S 1 S 1 Fially, we eed a formula from [Zv], expressig the measure of a sectio i terms of the Fourier trasform. This formula geeralizes the correspodig result for volume; see [4]. Propositio 4. ([Zv]) et be a origi-symmetric star body i R, the, for every ξ S 1, µ( ξ ) = 1 / x ( ) tx +1 t f dt (ξ), π where the Fourier trasform of the fuctio of x R i the right-had side is a cotiuous homogeeous of degree 1 fuctio o R \ {}. 3. Stability The followig elemetary fact was used by Zvavitch [Zv] i his geeralizatio of the Busema-Petty problem. emma 1. et a,b > ad let α be a o-egative fuctio o (, max{a,b}] so that the itegrals below coverge. The a a t 1 α(t)dt a t α(t)dt b b t 1 α(t)dt a t α(t)dt. (8)

7 HYPERPANE INEQUAITY FOR MEASURES 7 Proof : The iequality (8) is equivalet to b a a t α(t)dt b a t 1 α(t)dt. Note that the latter iequality also holds i the case a b. The measure of a body ca be expressed i polar coordiates as follows: x 1 µ() = f(u) du = t 1 f(tx)dt dx. (9) S 1 I particular, if f = 1 we get the polar formula for volume: Vol () = x dx. (1) S 1 We are ready to prove Theorem. Proof of Theorem. First, we rewrite the coditio (4) usig Propositio 4: x ( ) tx x +1 t f dt (ξ) x +1 x ( ) tx t f dt for each ξ S 1. By Propositio, the fuctio 1 (ξ) + πε (11) represets a positive defiite distributio. et µ be the measure o S 1 correspodig to this positive defiite distributio by Propositio 1. Itegratig (11) over S 1 with respect to the measure µ ad applyig the spherical Parseval formula, Propositio 3, we get S 1 x +1 x ( ) tx t f dt (ξ)dµ (ξ)

8 8 AEXANDER ODOBSY ad so S 1 x +1 x S 1 ( ) tx t f dt S 1 (ξ)dµ (ξ) + πε dµ (ξ), S 1 t f(tx)dt dx t f(tx)dt dx + πε dµ (ξ). S 1 (1) Applyig emma 1 with a =, b = x 1, α(t) = f(tx), we get S 1 S 1 t 1 f(tx) dt t 1 f(tx)dt t f(tx) dt x 1 t 1 f(tx)dt dx S 1 x 1 t 1 f(tx)dt dx S 1 t f(tx)dt, x S 1, t f(tx)dt dx (13) t f(tx)dt dx. Addig iequalities (1) ad (13) ad usig the polar formula (9), we get x 1 t 1 f(tx)dt dx S 1 S 1 t 1 f(tx)dt dx + πε dµ (ξ), S 1

9 HYPERPANE INEQUAITY FOR MEASURES 9 ad µ() µ() + πε dµ (ξ). S 1 It remais to estimate the itegral i the right-had side of the latter iequality. For this we use the formula for the Fourier trasform (i the sese of distributios; see [GS, p.194]): ( ) x +1 π +1 (ξ) = Γ( 1 ) ξ 1. Usig Parseval s formula agai, Propositio 3, ad Hölder s iequality, we obtai πε = πεγ( 1 ) S 1 dµ (ξ) = πεγ( 1 ) π +1 ( +1 S 1 ( ) (ξ)dµ (ξ) πεγ( 1 π +1 dx ) S 1 π +1 Now use the polar formula for volume (1) ad ote that πγ( 1 ) π +1 S 1 1 1/ = 1 c. S 1 x dx ) 1/ S 1 1. Stability i the origial Busema-Petty problem was studied i [5], where it was show that if the dimesio 4, the for ay origi-symmetric covex bodies ad i R ad every ε >, the iequalities imply Vol 1 ( ξ ) Vol 1 ( ξ ) + ε, Vol () 1 ξ S 1 Vol () 1 + c ε. (14) This is stroger tha what Theorem provides i the case of volume. I fact, if µ i Theorem is volume (f 1), the (5) reads as Vol () Vol () + ε 1 c Vol () 1/, which follows from (14) by the Mea Value Theorem applied to the fuctio h(t) = t /( 1). However, Theorem works for arbitrary measures, while the approach of [5] does ot allow this degree of geerality. Theorem does ot hold true i dimesios greater tha four, simply because the aswer to the Busema-Petty problem i these dimesios is egative. However, the statemet of this theorem becomes correct i all dimesios if we assume i additio that is a itersectio body (see [3, Ch.4] for the defiitio ad properties of

10 1 AEXANDER ODOBSY itersectio bodies). It was proved i [1] that a origi-symmetric star body i R is a itersectio body if ad oly if the fuctio 1 represets a positive defiite distributio. The proof of Theorem remais exactly the same i this case. Corollary 1 holds i all dimesios uder a additioal assumptio that ad are both itersectio bodies, ad the result of Theorem 1 is valid i all dimesios whe is a itersectio body (the uit Euclidea ball is a itersectio body, so the iequality (3) is sharp i the class of itersectio bodies i every dimesio). Note that Propositio meas that every origi-symmetric covex body i R, 4 is a itersectio body. The latter is o loger true if 5. Ackowledgemets. I wish to thak the US Natioal Sciece Foudatio for support through grats DMS ad DMS Part of the work was doe durig my stay at the Max Plack Istitute i Bo. I would like to thak the MPI for support ad hospitality. I thak the referee for valuable commets. Refereces [Ba]. Ball, Some remarks o the geometry of covex sets, Geometric aspects of fuctioal aalysis (1986/87), ecture Notes i Math. 1317, Spriger-Verlag, Berli-Heidelberg-New York, 1988, [Bo1] J. Bourgai, O high-dimesioal maximal fuctios associated to covex bodies, Amer. J. Math. 18 (1986), [Bo] J. Bourgai, Geometry of Baach spaces ad harmoic aalysis, Proceedigs of the Iteratioal Cogress of Mathematicias (Berkeley, Calif., 1986), Amer. Math. Soc., Providece, RI, 1987, [Bo3] J. Bourgai, O the distributio of polyomials o high-dimesioal covex sets, Geometric aspects of fuctioal aalysis, Israel semiar (19899), ecture Notes i Math Spriger, Berli, 1991, [Bo4] J. Bourgai, O the Busema-Petty problem for perturbatios of the ball, Geom. Fuct. Aal. 1 (1991), [BP] H. Busema ad C. M. Petty, Problems o covex bodies, Math. Scad. 4 (1956), [DP] N. Dafis ad G. Paouris, Small ball probability estimates, ψ -behavior ad the hyperplae cojecture, J. Fuct. Aal. 58 (1), [E] R. Elda ad B. lartag, Approximately gaussia margials ad the hyperplae cojecture, preprit, arxiv: [G1] R. J. Garder, Itersectio bodies ad the Busema-Petty problem, Tras. Amer. Math. Soc. 34 (1994), [G] R. J. Garder, A positive aswer to the Busema-Petty problem i three dimesios, A. of Math. () 14 (1994), [G3] R. J. Garder, Geometric tomography, Secod editio, Cambridge Uiversity Press, Cambridge, 6.

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