Communications in Mathematical Sciences

Volume 15 (2017)

Number 1

Extreme points of a ball about a measure with finite support

Pages: 77 – 96



Houman Owhadi (California Institute of Technology Pasadena, Calif., U.S.A.)

Clint Scovel (California Institute of Technology Pasadena, Calif., U.S.A.)


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.


extreme points, Prokhorov, Kantorovich, Monge–Wasserstein, Strassen, Kantorovich–Rubinstein, optimization, ambiguity

2010 Mathematics Subject Classification

52A05, 60D05

Published 10 January 2017