Here is a natural question which I have been unable to find discussed in the literature. If $K$ is a compact convex set in a locally convex topological vector space, is there a Choquet simplex $\Delta$ and a continuous affine map $f:\Delta\to K$ sending $\partial_e(\Delta)$ homeomorphically (or at least bijectively) to $\partial_e(K)$? If so, is it unique up to isomorphism over $K$? Is this at least true in the metrizable case? It would be natural to think of compact convex sets as being given by "generators" (extreme points) and "relations" with Choquet simplexes the "free" compact convex sets (although this maybe cannot be taken too literally).
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$\begingroup$ It is not hard to show the answer is yes if $\partial_e(K)$ is closed. The tricky case is when the set of extreme points of $K$ is not closed. $\endgroup$– Bruce BlackadarCommented May 28, 2023 at 18:29
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$\begingroup$ I have found a counterexample to the uniqueness statement in the case where $\partial_e(K)$ is not closed. Roughly speaking, let $K$ be a compact convex set with a sequence of extreme points, such that the convex hull of the first four is a square and the sequence of extreme points converges to the center of the square. There are many nonisomorphic Choquet simplexes which cover $K$. $\endgroup$– Bruce BlackadarCommented May 28, 2023 at 21:16
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