Extreme points of a ball about a measure with finite support

TL;DR

This paper characterizes the extreme points of certain metric balls around measures with finite support in Polish spaces, and develops linear programming methods for their efficient computation.

Contribution

It provides a precise description of extreme points for metric balls around finitely supported measures and introduces linear programming representations for their computation.

Findings

Extreme points have supports with at most n+2 points.

Linear programming representations enable efficient computation.

Results apply to Prokhorov, Monge-Wasserstein, and Kantorovich metric balls.

Abstract

We show that, for the space of Borel probability measures on a Borel subset of a Polish metric space, the extreme points of the Prokhorov, Monge-Wasserstein and Kantorovich metric balls about a measure whose support has at most n points, consist of measures whose supports have at most n+2 points. Moreover, we use the Strassen and Kantorovich-Rubinstein duality theorems to develop representations of supersets of the extreme points based on linear programming, and then develop these representations towards the goal of their efficient computation.