Logarithmically-concave moment measures I

TL;DR

This paper introduces a Riemannian metric linked to log-concave measures in R^n, used to bound derivatives and spectral gaps in solutions to the toric Kähler-Einstein equation, advancing geometric analysis of such measures.

Contribution

It presents a new metric associated with log-concave measures and applies it to derive bounds and spectral gap estimates for the toric Kähler-Einstein equation.

Findings

Bounded second derivatives of solutions.

Established spectral-gap estimates.

Linked geometric structures with measure properties.

Abstract

We discuss a certain Riemannian metric, related to the toric Kahler-Einstein equation, that is associated in a linearly-invariant manner with a given log-concave measure in R^n. We use this metric in order to bound the second derivatives of the solution to the toric Kahler-Einstein equation, and in order to obtain spectral-gap estimates similar to those of Payne and Weinberger.