对数凹函数熵的低维Busemann-Petty问题.pdf
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1、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
2、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 proble
3、m 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 i
4、n 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
5、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 a
6、nswer 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,B
7、OURGAIN 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
8、 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,JRnnHJRnnH
9、Ent(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 defini
10、tion 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 defin
11、ition 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
12、 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 fu
13、nction 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)
14、+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.P
15、roof 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
16、 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
17、,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
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