对数凹函数熵的低维Busemann-Petty问题.pdf

引入对数凹函数嫡的Busemann-Petty 问题，即对于2 个R上的偶的对数凹函数f和g,且f和g具有正的、有限的积分，假设 JrnH 于(a)da JenH 9(r)da 对于任意i维子空间 H 均成立，是否能够得到Ent(f）En t(g).得到了该问题的部分解答，为解决凸体上的低维Busemann-Petty问题提供了一种新的途径.关键词：低维Busemann-Petty问题；对数凹函数；i-相交体；i-相交函数；熵函数Received date:2022-09-29Foundation item:The National Natural Science Foundation of China(11701373);Shanghai Sailing Program(17YF1413800)Biography:MA Dan(1985),female,associate professor,research area:convex geometric analysis.E-mail:引用格式：马丹，王亚龙.对数凹函数熵的低维Busemann-Petty问题J.上海师范大学学报（自然科学版),2 0 2 3,52(3)：303-310.Citation format:MA D,WANG Y L.The lower dimensional Busemann-Petty problem on entropy of log-concavefunctions JJ.Journal of Shanghai Normal University(Natural Sciences),2023,52(3):303-310.3041IntroductionLet Vi()and V()denote the i-dimensional and the n-dimensional Lebesgue measure respectively,and let Gn,idenote the Grassmann manifold of i-dimensional linear subspaces in Rn.Let Kr denote the set of n-dimensionalconvex bodies containing the origin in their interiors,and let Sn-1 denote the Euclidean sphere.Throughout thispaper,we let 1 i n-1.The lower dimensional Busemann-Petty problem which also called the generalized Busemann-Petty problemasks:suppose that K and L are two origin-symmetric convex bodies in Rn so thatfor every H Gn,n-i.Does it followFor i=1,it is the celebrated Busemann-Petty problem.To solve this problem,in 1,LUTWAK introduced thenotion of the intersection body of a star body,and the problem has affirmative answer for 2 n 4 and hasnegative answer for n 5(see more references in 2-8).For 2 i n-1,in 9,BOURGAIN etc.proved thatthe problem has a negative answer for 4 i n-1.For i=2 and i=3,it has still been an open problem in thelast two decades,and has gained extensive attention.In some special cases,the problem has made breakthroughs.In 9,BOURGAIN etc.gave an affirmative answer for the case i=2,when L is a ball and K is close to L.In10,RUBIN gave an affirmative answer when the body with smaller sections is a body of revolution.However,the problem remains unsolved in a general situation.In order to better study this problem,ZHANG in 11 andKOLDOBSKY in 12 gave the definition of i-intersection body respectively.In 12,MILMAN pointed that twotypes of generalizations of the notion of intersection bodies are not equivalent.A function f:Rn 0,+oo)is log-concave if for every,y E IRn and 0 t 0.Ilallk=min(入0 :E 入K.305f()dc g(a)dc,JRnnHJRnnHEnt(f)Ent(g)?(2)306Letting :IRn-IR U+oo),is convex if for every a,y E IRn and t E(o,1),From the definition of log-concave function(1),every log-concave function f:IRn 0,+oo)has the formFor an integrable function f:Rn 0,+co)with f(o)0 and any p 0,in 19,BALL introduced the set of Kp(f),From the definition of radial function(2),we haveXPKp(f)(a)prpf(o)CIn 20,we get the following properties.Let f:IRn-0,+oo)be an integrable function with f(o)0.For everyp 0,we have that(i)o E Kp(f);(ii)Kp(f)is a star-shaped set;(ii)Kp(f)is symmetric if f is even;(iv)(Kn(f)=o Jen(a)da.From 12,KOLDOBSKY gave the definition of i-intersection bodies,that is,for two origin-symmetric starbodies K and L in Rn,K is an i-intersection body of Lif for every H e Gn,n-i such thatBy the polar formula for the volume,the above equality can be written as the formSo a star body K is an i-intersection body,if there exists a star body L such that K=In,iL.3Main results and proofsIn this section,we study the lower dimensional Busemann-Petty problem on the entropy of log-concave func-tions.First,we give the definition of i-intersection function.Definition 1 Let f:Rn 0,+oo)be a positive integrable function.The i-interscetion function In,if:IRn 0,+oo)is defined asf:IRn 0,+oo)with f(o)0 is an i-intersection function,if there exists a positive integral function g suchthat f(c)=In,ig(c).Now,we study the equivalence between i-intersection bodies and i-intersection functions.J.Shanghai Normal Univ.(Nat.Sci.)p(1-t)r+ty)(1-t)p(a)+tp(y).ERnJo1Vi(Kn H)=Vn-i(Ln H).Ilullkdu=sn-inHln-iJsn-1nHIll/+idu.In,if(a)=exp Jun.2023f=e-.8f(rc)rp-1drc)drPfor a E Rn(o.Vol.52,No.3Lemma 1 Let f:IRn 0,+oo)be a positive continuous integrable function with f(o)0.Then,f is ani-intersection function,if and only if Kn-i(f)is an i-intersection body.Proof If f is an i-intersection function,according to definition 1,there exists a positive continuous integrablefunction g with g(o)O such that MA D,WANG Y L:The Lower Dimensional Busemann-Petty Problem on.f(a)=In,ig(a)=(ex307Meanwhile,by the definition of Kn-i(g),we havenPKn-i(f)(a)=f(o)-It is clear that Kn-i(f)is an i-intersection body.Reversely,if Kn-i(f)is an i-intersection body,there exists an origin-symmetric star body L such thatSet g(c)=e-llL,g(o)=1.A direct calculation yields Kn-i(g)=I(n-i+1)-n L.Then,L=I(n-i+1)=-:Kn-i(g),and from the definition of i-intersection body,we haveKn-i(f)=In,L=In,(T(n-i+1)Kn-i(g)=I(n-i+1)In,i n-i(g).Thus,By the definition of Kn-i(f)and its radial function,we haveg(o)pIn,t Kn-i(g)(a).PKn-i(f)(a)=PIn,iL(ac).PKn-()(a)=I(n-i+1)-+pIn,n-(g)(a).8(rn-i-1 In,ig(ra)drIn.i Kn-i(g)(a).Therefore,PKn-(f)(a)=cpKn-(In.tg)(a),c=IBy 18,for fixed p O and t O,we have Kp(f)=tKp(g),if and only if f(c)=g(t-1).Therefore,f isan i-intersection function.Next,we give the following lemma to better understand the relationship between the integral of log-concavefunctions and the volume of star bodies.Lemma 2 Let K be an origin-symmetic star body in R,f(a)=e-ll,for a E IR,H e Gn,n-.Then,/Rnf()log f(a)dc=-nI(n+1)V(K).RnI(n-i+1)-1,f(c)dc=I(n-i+1)Vn-i(K n H),JRnnHf()dc=T(n+1)V(K),308Proof By polar coordinates and the volume formula,we havee-p-(K,a)daJRnnHJRnnH+8JoJsn-1nH=(n-i)Vn-i(Kn H)=I(n-i+1)Vn-i(K n H).J.Shanghai Normal Univ.(Nat.Sci.)(a)dc+80tn-i-le-tdtJoJun.2023rn-i-1e-p-1(K,ru)dudrSimilarly,we haveK,a)d.crn-1e-p-(K,ru)dudrn-1+80Rnandf(c)log f(c)da=RnWe also need the following geometric result.Lemma 311 Let K be an i-intersection body in Rn and let L be an origin-symmetric star body in IR satis-fyingfor every H Gn,n-i.Then,Now,we give the main results and their proofs.Theorem 3 The lower dimensional Busemann-Petty problem on the entropy of log-concave functions hasaffrmative answers for f(a)=e-lllk and g(a)=e-lll,where a e R,K is an i-intersection body in IRn andL is an origin-symmetric star body in Rn.Proof Let K be an i-intersection body in IRn and let L be an origin-symmetric star body in Rn,and let f()=e-llk and g(a)=e-ll for a E IRn.Since the function tlogt is increasing for t e-1l and Ent(cf)=cEnt(f)for a constant c,without loss of generality,we assume J(f)e-1 and J(g)e-1.Now,for an arbitrary H EGn,n-i,suppose JanH f(a)da JrnH g(a)da.By lemma this s equivalent to Vn-s(Kn H)Vn-i(Ln H)./Rn=nV(K)=I(n+1)V(K),K,p-1(K,a)daRnX0Sn-1=-nF(n+1)V(K).Vn-i(Kn H)Vn-i(Ln H),V(K)V(L).tn-le-tdtrn-le-p-(K,ru)p-1(K,ru)dudrVol.52,No.3By lemma,we have V(K)V(L).Thus,=Ent(g),which completes the proof.Finally,we further investigate the relationship between the original lower dimensional Busemann-Petty problemand its version on the entropy of log-concave functions,which might provide a new path to study this long-standingopen problem.Proof of theorem 2 Let K,L e Kr.Similar to 18,without loss of generality,we assume V(K)1 andV(L)1.Let H be an arbitrary(n-i)-dimensional subspace in Rn,and let f(a)=e-llk and g(a)=e-llL,for E Rn.By lemma 2,we haveSimilarly,Hence,MA D,WANG Y L:The Lower Dimensional Busemann-Petty Problem on.Ent(f)=/f(a)log f(a)da-J(f)log J(f)JRn=-n(n+1)V(K)-(n+1)V(K)log(T(n+1)V(K)-nI(n+1)V(L)-I(n+1)V(L)log(T(n+1)V(L)f(c)da=r(n-i+1)Vn-i(Kn H).JRnnHJ(f)=I(n+1)V(K),f(c)log f(a)da=-n(n+1)V(K)./RnEnt(f)=/Rn=-T(n+1)V(K)(n+log(n+1)log V(K).309f(a)og f(a)da-J(f)log J(f)Analogies holdforL.Now,we assume Vn-i(K n H)Vn-i(L n H)for every H e Gn,n-i,by lemma 2,which is equivalent tof(ac)daJRnnHJRnnHfunctions has affirmative answer,we have Ent(f)Ent(g),which implies V(K)V(L).Therefore,the lowerdimensional Busemann-Petty problem also has affirmative answer.g(c)da.If the lower dimensional Busemann-Petty problem on the entropy of log-concaveReferences:1 LUTWAK E.Intersection bodies and dual mixed volumes JJ.Advances in Mathematics,1988,71(2):232-261.2BUSEMANN H,PETTY C M.Problems on convex bodies J.Mathematica Scandinavica,1956,4:88-94.3 GARDNER R J.Intersection bodies and the Busemann-Petty problem J.Transactions of the American MathematicalSociety,1994,342(1):435-445.3104GARDNER R J,KOLDOBSKY A,SCHLUMPRECHTT.An analytic solution to the Busemann-Petty problem on sectionsof convex bodies J.Annals of Mathematics,1999,149(2):691-703.5GIANNOPOULOS A.A note on a problem of H.Busemann and C.M.Petty concerning sections of symmetric convexbodies J.Mathematika,1990,37(2):239-244.6KOLDOBSKY A.Intersection bodies in R4 J.Advances in Mathematics,1998,136(1):1-14.7GARDNER R J,ZHANG G.Centered bodies and dual mixed volumes J.Transactions of the American MathematicalSociety,1994,345(2):777-801.8ZHANG G.A positive solution to the Busemann-Petty problem in R4 J.Annals of Mathematics,1999,149(2):535-543.9BOURGAIN J,ZHANG G.On a Generalization of the Busemann-Petty Problem CJ/Convex Geometric Analysis,volume 34 of Mathematical Sciences Research Institute Publications.Cambridge:Cambridge University Press,1999:65-76.10 RUBIN B.The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry J.IsraelJournal of Mathematics,2009,173:213-233.11 ZHANG G.Sections of convex bodies J.American Journal of Mathematics,1996,118(2):319-340.12 KOLDOBSKY A.Fourier Analysis in Convex Geometry M.Providence,RI:American Mathematical Society,2005.13 MILMAN E.Generalized intersection bodies are not equivalent J.Advances in Mathematics,2008,217(6):2822-2840.14 COLESANTI A.Brunn-Minkowski inequalities for variational functionals and related problems JJ.Advances in Mathe-matics,2005,194(1):105-140.15 COLESANTI A,FRAGALA I.The first variation of the total mass of log-concave functions and related inequalities J.Advances in Mathematics,2013,244:708-749.16 FANG N,ZHOU J.LYZ ellipsoid and Petty projection body for log-concave functions J.Advances in Mathematics,2018,340:914959.17LV S.A functional Busemann intersection inequality J.Journal of Geometric Analysis,2021,31(6):6274-6291.18 FANG N,ZHOU J.The Busemann-Petty problem on entropy of log-concave functions J.Science China Mathematics,2022,65(10):2171-2182.19 BALL K.Logarithmically concave functions and sections of convex sets in R JJ.Studia Mathematica,1988,88(1):69-84.20 ARTSTEIN-AVIDAN S,GIANNOPOULOS A,MILMAN V D.Asymptotic Geometric Analysis,Part I M.Providence,RI:American Mathematical Society,2015.(责任编辑：冯珍珍)J.Shanghai Normal Univ.(Nat.Sci.)Jun.2023