Cone-volume measure and stability

TL;DR

This paper proves that the cone-volume measure of convex bodies with centroid at the origin satisfies a key concentration condition, leading to a best-possible inequality for the U-functional, with stability versions included.

Contribution

It establishes the subspace concentration condition for cone-volume measure and derives a sharp inequality for the U-functional, with improved stability results.

Findings

Cone-volume measure satisfies subspace concentration condition.

Derived the best possible inequality for the U-functional.

Provided stability versions of the main inequalities.

Abstract

We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the $\mathrm{U}$-functional of a convex body. For both results we provide stronger versions in the sense of stability inequalities.