Cone-volume measures of polytopes

TL;DR

This paper proves a key property of cone-volume measures for polytopes with centroid at the origin, leading to the complete proof of a long-standing affine isoperimetric inequality for the U-functional and its equality conditions.

Contribution

It establishes the subspace concentration condition for cone-volume measures and fully proves a conjectured affine isoperimetric inequality for the U-functional.

Findings

Cone-volume measure satisfies the subspace concentration condition.

Complete proof of the affine isoperimetric inequality for the U-functional.

Identification of equality conditions for the inequality.

Abstract

The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the U-functional is completely established -- along with its equality conditions.