Affiliation:
1. Department of Mathematics, University of California, Los Angeles, CA 90095
Abstract
Abstract
We study the inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if $K, L$ are convex bodies such that the volume of the projection of $K$ onto any hyperplane through the origin is bounded by the volume of the intersection of $L$ with the hyperplane, then $|K| \le |L|$. Firstly, we study the reverse question: in particular, we show that if $K, L$ are origin-symmetric convex bodies in John’s position such that the volume of the intersection of $K$ with any hyperplane through the origin is bounded by the volume of the projection of $L$ onto the hyperplane, then $|K| \le \sqrt {n}|L|$. The condition we consider is weaker than the conditions that appear in the Busemann–Petty and Shephard problems. Secondly, we appropriately extend the result of Giannopoulos and Koldobsky to various classes of measures possessing concavity properties, including log-concave measures.
Publisher
Oxford University Press (OUP)
Reference33 articles.
1. Shadows of convex bodies;Ball;Trans. Amer. Math. Soc,1991
2. Volume ratios and a reverse isoperimetric inequality;Ball;J. Lond. Math. Soc. (2),1991
3. Convex set functions in $d$-space;Borell;Period. Math. Hungar.,1975
4. On the Distribution of Polynomials on High-Dimensional Convex Sets;Bourgain;Geometric Aspects of Functional Analysis 1469 in Lecture Notes in Math,1991
Cited by
7 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献