Pseudo-cones and measure transport
Rolf Schneider
TL;DR
This work interprets the Gauss image problem for $C$-pseudo-cones as a measure transport problem implemented by the reverse radial Gauss map $\alpha_K^*$. A natural cost $c(u,v)=\log|\langle u,v\rangle|$ is identified, and the corresponding transport is shown to be optimal in the sense of minimizing the total cost, paralleling the Brenier–McCann framework but in the pseudo-cone setting. A Rockafellar-type criterion is established for pseudo-cone subdifferentials: a set is $c$-cyclically monotone iff it embeds into the pseudo-subdifferential $\partial^{\bullet}K$ of some $C$-pseudo-cone. The results thus connect measure transport, convex-analytic duality, and the geometry of pseudo-cones, extending classical subdifferential theory to this conic setting.
Abstract
A recent result on the Gauss image problem for pseudo-cones can be interpreted as a measure transport, performed by the reverse radial Gauss map of a pseudo-cone. We find a cost function that is minimized by this transport map, and we prove an analogue of Rockafellar's characterization of the subdifferentials of convex functions.