Remarks on mass transportation minimizing expectation of a minimum of affine functions

TL;DR

This paper analyzes a specific optimal transport problem with a cost function defined as the minimum of several affine functions, showing it reduces to a finite-dimensional extremal problem with solutions concentrated on particular partitions.

Contribution

It establishes an equivalence between the Monge--Kantorovich problem with this cost and a finite-dimensional extremal problem, extending to multiple marginals.

Findings

Solution concentrated on unions of product sets with specific properties

Partitions of the real line solve an auxiliary extremal problem

Partial generalization to more than two marginals

Abstract

We study the Monge--Kantorovich problem with one-dimensional marginals $\mu$ and $\nu$ and the cost function $c = \min\{l_1, \ldots, l_n\}$ that equals the minimum of a finite number $n$ of affine functions $l_i$ satisfying certain non-degeneracy assumptions. We prove that the problem is equivalent to a finite-dimensional extremal problem. More precisely, it is shown that the solution is concentrated on the union of $n$ products $I_i \times J_i$, where $\{I_i\}$ and $\{J_i\}$ are partitions of the real line into unions of disjoint connected sets. The families of sets $\{I_i\}$ and $\{J_i\}$ have the following properties: 1) $c=l_i$ on $I_i \times J_i$, 2) $\{I_i\}, \{J_i\}$ is a couple of partitions solving an auxiliary $n$-dimensional extremal problem. The result is partially generalized to the case of more than two marginals.