Stability of the cone-volume measure with near constant density

TL;DR

This paper proves that convex bodies with cone-volume measures nearly constant in density are close to the unit ball in shape, establishing a stability result in convex geometry.

Contribution

It introduces a stability theorem linking near-constant cone-volume measure density to the shape proximity of convex bodies to the sphere.

Findings

Convex bodies with nearly constant cone-volume measure are close to the sphere in $L^2$-distance.

The result provides a quantitative stability estimate.

The proof connects measure density with geometric proximity.

Abstract

We prove that if the density of the cone-volume measure of a smooth, strictly convex body with respect to the spherical Lebesgue measure is nearly constant, then a homothetic copy of the body is close to the unit ball in the $L^2$-distance.