  We are interested in the cost-convex potentials in optimal mass transport
theory, and we show by direct and geometric arguments the equivalence between
cost-subdifferentials and ordinary subdifferentials of cost-convex functions,
under the assumptions A0, A1, A2, and A3W on cost functions introduced by Ma,
Trudinger, and Wang. The connectivity of contact sets of optimal transport maps
follows as a direct corollary. Our approach is quite different from the
previous result of Loeper which he obtained as the first step toward his
Hoelder regularity theory of potential functions, and which was based upon
approximation using the regularity theory of Ma, Trudinger, and Wang. The
result in this paper improves his result, by relaxing certain geometrical
assumptions on the domains and targets; it also completes his Hoelder
regularity theory of potential functions on the round sphere, by making it
self-contained.
