I was curious if an analogue of the open mapping theorem existed for affine maps between compact convex spaces. I'm interested in a question like the following:
Suppose that $\mathcal{K}, \mathcal{L}$ are both Choquet simplices, and that $f : \mathcal{K} \twoheadrightarrow \mathcal{L}$ is a continuous, affine, surjective map which maps boundary points of $\mathcal{K}$ to boundary points of $\mathcal{L}$. Does it follow that $f$ is open?
My first thought was to try to see if the textbook proof of the open mapping theorem could be applied by expanding my simplices into vector spaces $\overline{\operatorname{span}}(\mathcal{K}), \overline{\operatorname{span}}(\mathcal{L})$, but this had several issues. For one, the (linear) open mapping theorem as far as I can tell from textbooks requires that the underlying space be metrizable, which is not the case if, say $\mathcal{K}$ is a space of probability measures with the weak* topology, since the weak* topology is only metrizable on norm-bounded subsets. But there was also the issue of how to deal with what happens near the boundary. For example, even in the case where $\mathcal{K}$ is a finite-dimensional simplex, things get a little more complicated if I move to a part of the simplex that isn't locally Euclidean.
My hope is that some of my hypotheses above are a little overkill. I just wanted to start with the best-case scenario, where $\mathcal{K}$ and $\mathcal{L}$ are the nicest possible compact convex spaces (Choquet simplices) and $f$ is a nice map between them.
Any help or references are appreciated, as are any counterexamples.