Existence of differentiable curves in convex sets and the concept of direction of the flow in mass transportation

TL;DR

This paper characterizes when differentiable curves exist within convex sets in locally-convex topological vector spaces and applies this to the mass transportation problem, using Colombeau algebras to define directions.

Contribution

It provides a new criterion for the existence of differentiable curves in convex sets and extends the mass transport equation to a generalized setting with Colombeau algebras.

Findings

Existence of curves characterized by tangent cone closure

Application to weakly differentiable probability measure families

Generalized mass transport equation framework

Abstract

In this paper we consider convex subsets of locally-convex topological vector spaces. Given a fixed point in such a convex subset, we show that there exists a curve completely contained in the convex subset and leaving the point in a given direction if and only if the direction vector is contained in the sequential closure of the tangent cone at that point. We apply this result to the characterization of the existence of weakly differentiable families of probability measures on a smooth manifold and of the distributions that can arise as their derivatives. This gives us a way to consider the mass transport equation in a very general context, in which the notion of direction turns out to be given by an element of a Colombeau algebra.