A transport inequality on the sphere obtained by mass transport
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.
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