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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?

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