Regularity of optimal transport between planar convex domains (Q2178468)

The main goal of this paper is to prove that in the plane one can go beyond \(C^\delta\)-regularity without further assumption on the domains other than convexity. More precisely, the authors establish a global \(W^{2,p}\)-estimate for potentials of optimal transport maps between convex domains in the plane. The proofs combine tools related with obliqueness in general convex domains with related estimates for the growth of eccentricity of sections of the potentials. The paper is very well written and it opens large perspectives in the regularity theory of various classes of minimization problems.