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

Abstract

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