It is a theorem of Hervé that
A compact convex set $K$ admits a strictly convex and continuous real function only if $K$ is metrizable. (The converse is also true.)
I'm wondering if any results of this sort are known if continuity is replaced with subdifferentiability. That is,
What topological constraints must a compact convex set $K$ meet in order to admit a strictly convex function that has a subgradient at every point in $K$? Does $K$ have to be first countable, for instance?