On Strassen's theorem for support functions

Strassen [23] established that there exists a two step martingale with marginal distributions mu, nu if and only if mu, nu are in convex order. Recently Chone, Gozlan and Kramarz [6] obtained a transport characterization of the stochastic order defined by convex positively 1-homogeneous functions, in the spirit of Strassen's theorem under certain technical assumptions. In this note we prove the result of [6] in full generality. We also observe that the restriction of the result to the case where mu, nu are supported on a half space is equivalent to Strassen's classical theorem.