An affirmative solution to the generalized Busemann--Petty problem with subspace dimensions $2$ and $3$

Abstract

The generalized Busemann--Petty problem asks whether origin-symmetric convex bodies in ${\mathbb{R}}^n$ having larger volume of all $m$-dimensional sections necessarily have larger volume. When $m\ge 4$, this is known to be false, but the cases $m=2,3$ for $n\ge 5$ have remained open since the 1990s. In this paper, we resolve these cases. Together with the known results, the generalized Busemann--Petty problem is completely solved: the answer is affirmative for $m=1,2,3$, and negative for $m\ge 4$.

12 pages