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

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 $m\geq 4$, this is known to be false, but the cases $m=2, 3$ for $n\geq 5$ have remained open since the 1990s. In this work, 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 $m\geq 4$. This talk is based on the joint work with Yu-de LIU and Ge XIONG.