# Extreme points of set of measures with given barycenter

Let $$X$$ be a convex compact metrizable subspace of a locally convex Hausdorff topological vector space, $$x_0\in X$$, and $$P$$ be the space of all Borel probability measures on $$X$$ with barycenter $$x_0$$.

Question: What are the extreme points of $$P$$?

In the special case of $$p\in P$$ whose support has finite-dimensional span, it's easy to deduce that $$p$$ is extreme if and only if its support is affinely independent. This, in particular, fully characterizes extreme points if $$X$$ is finite-dimensional.

I'm wondering whether there's a similarly simple/geometric characterization for the general case.