Borell's formula for a Riemannian manifold and applications

TL;DR

This paper extends Borell's formula from Gaussian vectors to Riemannian manifolds and demonstrates its applications, including a novel proof of a convolution inequality on the sphere, enriching the mathematical tools for geometric analysis.

Contribution

The authors generalize Borell's formula to Riemannian manifolds and apply it to derive new results like a convolution inequality on the sphere.

Findings

Extended Borell's formula to Riemannian manifolds

Provided a new proof of a sphere convolution inequality

Enhanced tools for geometric analysis on manifolds

Abstract

Borell's formula is a stochastic variational formula for the log-Laplace transform of a function of a Gaussian vector. We establish an extension of this to the Riemannian setting and give a couple of applications, including a new proof of a convolution inequality on the sphere due to Carlen, Lieb and Loss.