Extreme points of a ball about a measure with finite support

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 efficiently computable supersets of the extreme points.