Abstract
We study the optimal transportation mapping$\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$.Following a classicalapproach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the metric-measure space$M=(\mathbb{R}^d, g, \mu)$. We prove, in particular, that $M$ admits a non-negative Bakry--Émery tensor provided both $V$ and $W$ are convex.If the target measure $\nu$ is the Lebesgue measure on a convex set $\Omega$ and $\mu$ is log-concave we prove that $M$ is a $CD(K,N)$ space.Applications of these results include some global dimension-free a priori estimates of $\| D^2 \Phi\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration argumentswe proof some diameter estimates for $M$.
Highlights
This paper is motivated by the following problem
We are interested in efficient estimates of the Lipschitz constant supRd D2Φe · (D2Φ) or the integral
A classical example is given by a Caffarelli’s contraction theorem. According to this result every optimal transportation mapping ∇Φ pushing forward the standard Gaussian measure onto a log-concave measure ν with the uniformly convex potential W (i.e. D2W ≥ K · Id with K > 0)
Summary
This paper is motivated by the following problem. Given two probability measures μ = e−V dx and ν = e−W dx on Rd let us consider the optimal transportation mapping T = ∇Φ of μ onto ν and the associated Monge–Ampere equation e−V = e−W (∇Φ) det D2Φ. A classical example is given by a Caffarelli’s contraction theorem According to this result every optimal transportation mapping ∇Φ pushing forward the standard Gaussian measure onto a log-concave measure ν with the uniformly convex potential W (i.e. D2W ≥ K · Id with K > 0). Monge–Ampere equation, Hessian manifolds, metric-measure space, Bakry–Emery tensor, Sobolev spaces, log-concave measures, convex geometry. Applications of the Hessian structures in convex geometry can be found in a recent paper of Bo’az Klartag and Rohen Eldan [11] It was shown in [11] that the positive solution to the thin shell conjecture implies the positive solution to the slicing problem.
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