Transportation on spheres via an entropy formula

Gordon Blower

Published 2022-07-13Version 1

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.