On the Hessian of the optimal transport potential

Abstract

We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove BrascampLieb inequalities and thus prove a new generalised Reverse Brascamp-Lieb inequality. Mathematics Subject Classification (2000): 49Q20 (primary); 52A40, 44A35 (secondary).