  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 by 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 computations we obtain this result
by applying the tensorial maximum principle to the weighted Laplacian of the
Bakry-{\'E}mery 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.
