When do convexity and lower semicontinuity imply continuity? Let $X$ be a nonempty compact convex subset of a locally convex space. Say $X$ has the convex function property if every convex, lower semicontinuous $f:X\to\mathbb R$ is also continuous.
Question: Which $X$ have the convex function property?


*

*We know some $X$ do.  For instance, this is clearly true if $X$ is one-dimensional because every convex function on $[0,1]$ is continuous on $(0,1)$ and so (being convex) is upper semicontinuous.  More generally, using continuity on the relative interior and inducting on dimension, any $X$ with finitely many extreme points has the convex function property.

*We know some $X$ don't.  Indeed, an answer to this question https://math.stackexchange.com/questions/772841/convex-closed-and-unclosed-functions-and-lower-semicontinuity can be restricted to $X=\{(a,b)\in[0,1]^2:\ b\geq a^2\}$ for an example.


Is there a general characterization?
 A: Let me resolve a special case of the question, by leveraging the examples mentioned above.  Suppose $(K,\rho)$ is a compact metric space and $X$ is the set of Borel probability measures on $K$ endowed with its weak* topology.  I claim that $X$ has the convex function property if and only if $K$ is finite.  
Theorem 10.2 in "Convex Analysis" by Rockafellar implies that any convex function defined on a finite-dimensional simplex is upper semicontinuous.  This gives one direction.
Conversely, suppose $K$ is infinite.  Letting $k_\infty$ be an accumulation point of $K$ (which exists because $K$ is an infinite compact metrizable space), define the affine continuous function $\varphi:X\to\mathbb R^2$ given by $\varphi(x):=\int_K \left(\rho(k,k_\infty), \rho(k,k_\infty)^2\right)\text{ d}x(k)$.  Then, the convex function $f:X\to\mathbb R$ which takes $f(\delta_{k_\infty}):=0$ and $f(x):=\tfrac{\varphi_1(x)^2}{\varphi_2(x)}$ for every $x\neq\delta_{k_\infty}$ should work.  A failure of continuity is witnessed along sequence $(\delta_{k_n})_{n=1}^\infty$ where, $\{{k_n}\}_{n=1}^\infty\subseteq K\setminus\{k_\infty\}$ is a sequence converging to $k_\infty$.
A natural conjecture is now that a general $X$ will have the convex function property if and only if $X$ has finitely many extreme points.  Is this true?
