The Polar Cone of the set of monotone maps

TL;DR

This paper characterizes the polar cone of monotone transport maps, showing that each element can be represented as the divergence of a measure field with positive definite matrix values.

Contribution

It provides a novel representation of the polar cone of monotone maps using divergence of measure fields, advancing the theoretical understanding of monotone transport.

Findings

Every element of the polar cone can be represented as a divergence of a measure field.

The measure field takes values in positive definite matrices.

This representation offers new insights into the structure of monotone transport maps.

Abstract

We prove that every element of the polar cone to the closed convex cone of monotone transport maps can be represented as the divergence of a measure field taking values in the positive definite matrices.