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Remarks on curvature in the transportation metric

Klartag B., Kolesnikov A.
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 K\"ahler-Einstein equation $e^{\Phi} = \det D^2 \Phi$ on proper convex cones. We prove a generalization of this theorem showing that for every $\Phi$ solving this equation on a proper convex domain $\Omega$ the corresponding metric measure space $(D^2 \Phi, e^{\Phi}dx)$ has a non-positive Bakry-{\'E}mery tensor. Modifying the Calabi's computations we obtain this result by applying tensorial maximum principle to the weighted Laplacian of the Bakry-{\'E}mery tensor. All of the computations are carried out in the generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of probability measures we prove a third-order uniform dimension-free a priori estimate in spirit of the second-order Caffarelli's contraction theorem.