Variants of the Busemann–Petty Problem and of the Shephard Problem

Abstract

We provide an affirmative answer to a variant of the Busemann-Petty problem, proposed by V. Milman: Let K be a convex body in R and let D be a compact subset of R such that, for some 1 6 k 6 n− 1, |PF (K)| 6 |D ∩ F | for all F ∈ Gn,k, where PF (K) is the orthogonal projection of K onto F and D ∩ F is the intersection of D with F . Then, |K| 6 |D|. We also provide estimates for the lower dimensional Busemann-Petty and Shephard problems, and we prove separation in the original Busemann-Petty problem.