A transport inequality on the sphere obtained by mass transport
type: Item
instance of: scholarly article
title: A transport inequality on the sphere obtained by mass transport
zbMATH Open document ID: 1296.49039
published in: Pacific Journal of Mathematics
publication date: 2014-07-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1406.7721
review text: If \((M,g)\) is an \(n\)-dimensional Riemannian manifold with positive Ricci curvature satisfying \(\text{Ric}\geq (n-1)g\) and \(\sigma\) is its normalized Riemannian volume, then \(d\sigma=d\,\text{vol}/\text{vol}(M)\) is the Riemannian volume measure normalized to be a probability measure. For a probability density \(f\) on \(M\) with \(\int f d\sigma=1\), let \[ H_{n,\sigma}(f)=n\int(f-f^{1-1/n})d\sigma \] be a nonnegative convex functional of \(f\) representing the classical dimensional entropy. For a function \(c:[0,\pi]\to \mathbb R^+\), the associated Kantorovich transportation cost between two Borel probability measures \(\mu\) and \(\nu\) on \(M\) given by the distance \(d\) is defined by \[ {\mathcal W}_c(\mu,\nu)=\inf\limits_\pi\iint c(d(x,y))d\pi(x,y), \] where the infimum is taken over all probability measures \(\pi\) on \(M\times M\) projecting on \(\mu\) and \(\nu\), respectively. In this paper, making use of McCann's transportation map, the author establishes a transport inequality on compact manifolds with positive Ricci curvature. It is shown that for every probability density \(f\) on \(M\) the following inequality holds: \({\mathcal W}_{c_n}(f d\sigma,\sigma)\leq H_{n,\sigma}(f)\). Also, the author gives some properties of the cost \(c_n\) and explains how to derive the sharp spectral gap inequality.
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 6323660
zbMATH Keywords: transport inequality
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Ricci curvature
zbMATH Keywords: optimal mass transport
author: D. Cordero-Erausquin
MaRDI profile type: Publication
OpenAlex ID: W1992440578
arXiv ID: 1406.7721
DOI: 10.2140/PJM.2014.268.23
description: scientific article