On smoothness properties of optimal value functions at the boundary of their domain under complete convexity

This article studies continuity and directional differentiability properties of optimal value functions, in particular at boundary points of their domain. We extend and complement standard continuity results from In particular, we present sufficient conditions for the inner semicontinuity of feasible set mappings and, using techniques from nonsmooth analysis, provide functional descriptions of tangent cones to the domain of the optimal value function. The latter makes the stated directional differentiability results accessible for practical applications

A Generalization of Caffarelli's Contraction Theorem via (reverse) Heat Flow

A theorem of L. Caffarelli implies the existence of a map pushing forward a source Gaussian measure to a target measure which is more log-concave than the source one, which contracts Euclidean distance (in fact, Caffarelli showed that the optimal-transport Brenier map $T_{opt}$ is a contraction in this case). We generalize this result to more general source and target measures, using a condition on the third derivative of the potential, using two different proofs. The first uses a map $T$, whose inverse is constructed as a flow along an advection field associated to an appropriate heat-diffusion process. The contraction property is then reduced to showing that log-concavity is preserved along the corresponding diffusion semi-group, by using a maximum principle for parabolic PDE. In particular, Caffarelli's original result immediately follows by using the Ornstein-Uhlenbeck process and the Pr\'ekopa--Leindler Theorem. The second uses the map $T_{opt}$ by generalizing Caffarelli's argument, employing in addition further results of Caffarelli. As applications, we obtain new correlation and isoperimetric inequalities.Comment: 33 pages; corrected typos, shortened Section 6 and some of the standard proofs. To appear in Math. Anna