  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 $C^1$ 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.
