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?

interiorof its effective domain -- so if $X$ is not open, you also don't have the "convex function property". (This is of course the counterexample you gave in your comment.) $\endgroup$2more comments