Remarks on curvature in the transportation metric

Abstract

Abstract. According to a classical result of E. Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the "hyperbolic" toric Kähler-Einstein equation e Φ = det D 2 Φ on proper convex cones. We prove a generalization of this theorem by showing that for every Φ solving this equation on a proper convex domain Ω the corresponding metric measure space (D 2 Φ, e Φ dx) has a non-positive Bakry-Émery tensor. Modifying the Calabi computations we obtain this result by applying the tensorial maximum principle to the weighted Laplacian of the Bakry-Émery tensor. Our computations are carried out in a generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of the log-concave probability measures we prove a third-order uniform dimension-free apriori estimate in the spirit of the second-order Caffarelli contraction theorem, which has numerous applications in probability theory.

Moment measures and stability for Gaussian inequalities

“…Certain stability estimates can be obtained under (one-sided) uniform bounds on the Hessian of the logarithmic potential − log g. The proof is based on the Caffarelli contraction theorem (see [20] and the references therein, some new developments for higher order derivatives can be found in [18]). for some constant ε > 0.…”

An Optimal Transport Approach to Monge–Ampère Equations on Compact Hessian Manifolds

“…We wish to point out that in contrast to the complex setting solutions to (9) do not define Einstein metrics, in the sense that (9) is not a reformulation of the Einstein equation Ricg = λg. However, as mentioned above the geometric properties of solutions to equation (9) have very recently been studied by Klartag and Kolesnikov in [12]. Moreover, when M = R n , (9) has been studied as a twisted Kähler-Einstein equation on a corresponding toric manifold (see [21,2]) and when M is the real torus with the standard affine structure (9) has been studied as an analog of a twisted singular Kähler-Einstein equation in [11].…”

“…Finally, we remark that the local geometry of smooth measured metric spaces of the form (M, ∇dφ, µ) where φ and µ are related as in Theorem 2, have recently been studied by Klartag and Kolesnikov in [12]. It is interesting to note that our approach shows that a pair of measures (µ, ν) with smooth densities on M and M * determines a pair of measured metric spaces (M, ∇dφ, µ) and (M * , ∇ * dφ * , ν) of the form studied in [12] related by Legendre transform.…”

The Deficit in the Gaussian Log-Sobolev Inequality and Inverse Santaló Inequalities

“…The converse is also true: if ν is the moment measure of a given log-concave probability measure η o with a regular density as above, then the function η Þ Ñ T pν, ηq`Hpη|Lebq reaches its infimum at η o . Let us mention that the notion of moment measures together with the above characterization recently found several applications in convex geometry [41,42], probability theory [22,44] or functional inequalities [25]. Here, we will use this description of moment measures to reparametrize the inequality (10) in terms of η 1 , η 2 instead of ν 1 , ν 2 , yielding to the following equivalent statement: for all log-concave probability measures η 1 , η 2 with an essentially continuous log-density, it holds…”