A geometric approach to apriori estimates for optimal transport maps

A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma–Trudinger–Wang condition is the Pogorelov second-derivative bound. This translates to an apriori interior <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> C¹ estimate for smooth optimal maps. Here we give a new derivation of this estimate which relies in part on Kim, McCann and Warren’s observation that the graph of an optimal map becomes a volume maximizing spacelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost.