On semi-discrete sub-partitions of vector-valued measures

TL;DR

This paper introduces a new optimal transport framework for vector-valued measures, focusing on the semi-discrete case and highlighting fundamental differences from scalar cases, including potential non-existence of dual solutions.

Contribution

It presents the concept of optimal transport for vector-valued measures and explores the semi-discrete case, revealing key differences from scalar measures.

Findings

Differences between scalar and vector optimal transport cases

Possibility of non-existence of dual solutions in certain vector cases

Fundamental properties of semi-discrete vector measures

Abstract

We introduce a concept of optimal transport for vector-valued measures and its dual formulation. In this note we concentrate on the semi-discrete case and show some fundamental differences between the scalar and vector cases. A manifestation of this difference is the possibility of non-existence of optimal solution for the dual problem for feasible primer problems.