Transportation on spheres via an entropy formula

TL;DR

This paper establishes transportation inequalities on spheres using an entropy formula, linking relative entropy and Wasserstein distances, with implications for Riemannian manifolds and geometric analysis.

Contribution

It introduces a new entropy formula applicable to spheres and smooth manifolds, leading to transportation inequalities involving geodesic distances.

Findings

Proves transportation inequalities for measures on spheres with power cost functions.

Derives an explicit entropy formula valid on smooth Riemannian manifolds.

Connects entropy, curvature, and Wasserstein metrics in geometric contexts.

Abstract

The paper proves transportation inequalities for probability measures on spheres for the Wasserstein metrics with respect to cost functions that are powers of the geodesic distance. Let $\mu$ be a probability measure on the sphere ${\bf S}^n$ of the form $d\mu =e^{-U(x)}dx$ where $dx$ is the rotation invariant probability measure, and $(n-1)I+{\hbox{Hess}}\,U\geq {\kappa_U}I$, where $\kappa_U>0$. Then any probability measure $\nu$ of finite relative entropy with respect to $\mu$ satisfies ${\hbox{Ent}}(\nu\mid\mu) \geq (\kappa_U/2)W_2(\nu, \mu )^2$. The proof uses an explicit formula for the relative entropy which is also valid on connected and compact $C^\infty$ smooth Riemannian manifolds without boundary. A variation of this entropy formula gives the Lichn\'erowicz integral.