A note on the Busemann-Petty problem for bodies of certain invariance

TL;DR

This paper investigates the Busemann-Petty problem for a special class of convex bodies with coordinate invariance, extending known results to new symmetric cases in higher dimensions.

Contribution

It introduces and analyzes a class of convex bodies with specific invariance properties, providing new insights into the Busemann-Petty problem for these bodies.

Findings

Affirmative answer for the Busemann-Petty problem in certain invariant bodies

Extension of the problem to bodies with coordinate invariance

New results for convex bodies in higher dimensions

Abstract

The Busemann-Petty problem asks whether origin symmetric convex bodies in $\R^n$ with smaller hyperplane sections necessarily have smaller volume. The answer is affirmative if $n\leq 3$ and negative if $n\geq 4.$ We consider a class of convex bodies that have a certain invariance property with respect to their ordered k-tuples of coordinates in $\R^{kn}$ and prove the corresponding problem.