Pseudo-cones and measure transport

TL;DR

This paper explores measure transport via the reverse radial Gauss map of pseudo-cones, identifying a cost function minimized by this map and establishing a convex analysis analogue.

Contribution

It introduces a new measure transport interpretation for pseudo-cones and characterizes the subdifferentials of convex functions in this context.

Findings

Identified a cost function minimized by the reverse radial Gauss map.

Proved an analogue of Rockafellar's subdifferential characterization.

Connected measure transport with convex analysis in pseudo-cones.

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.