Variants of the Busemann-Petty problem and of the Shephard problem

TL;DR

This paper affirms a variant of the Busemann-Petty problem, providing new insights and estimates for related geometric problems involving convex bodies, projections, and intersections in Euclidean space.

Contribution

It offers an affirmative answer to a proposed variant of the Busemann-Petty problem and establishes estimates for lower-dimensional versions and separation results.

Findings

Confirmed a variant of the Busemann-Petty problem.

Provided estimates for lower-dimensional Busemann-Petty and Shephard problems.

Proved separation in the original Busemann-Petty 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 ${\mathbb R}^n$ and let $D$ be a compact subset of ${\mathbb R}^n$ such that, for some $1\ls k\ls n-1$, $$|P_F(K)|\ls |D\cap F|$$ for all $F\in G_{n,k}$, where $P_F(K)$ is the orthogonal projection of $K$ onto $F$ and $D\cap F$ is the intersection of $D$ with $F$. Then, $$|K|\ls |D|.$$ We also provide estimates for the lower dimensional Busemann-Petty and Shephard problems, and we prove separation in the original Busemann-Petty problem.