The Busemann–Petty problem via spherical harmonics

Abstract

The Busemann–Petty problem asks whether symmetric convex bodies in R n with smaller central hyperplane sections necessarily have smaller n-dimensional volume. The solution has recently been completed, and the answer is affirmative if n⩽4 and negative if n⩾5. In this article we present a short proof of the affirmative result and its generalization using the Funk–Hecke formula for spherical harmonics.