On Strassen's Theorem for support functions

Strassen established that there exists a two step martingale with marginal distributions $\mu$, $\nu$ if and only if $\mu$, $\nu$ are in convex order. Recently Choné-Gozlan-Kramarz 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 Choné-Gozlan-Kramarz result 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.