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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.

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