# A convex function is "usually" subdifferentiable

Let $$X$$ be a locally convex topological vector space, and let $$f:X\to\mathbb R\cup\{\infty\}$$ be a proper, convex, lower semicontinuous function, whose effective domain $$D:=f^{-1}(\mathbb R)$$ is compact. Let $$S\subseteq D$$ be the set of points at which $$f$$ is subdifferentiable.

My question is somewhat vague: What results are there that say $$S$$ constitutes "most of" $$D$$? One example of such a result is the Brøndsted-Rockafellar theorem, which says $$S$$ is dense in $$D$$ whenever $$X$$ is a Banach space.

Ideally, I'd like to know any results guaranteeing $$S$$ is most of $$D$$ in some sense other than topological (e.g. measure-theoretic). Any pointers to such results would be very helpful.

• I don't have an answer, but I can suggest some keywords in connection to your last paragraph. One generalisation of measure zero sets to more general vector spaces is given by sets called shy; a set that isn't shy is called prevalent. Commented Mar 30, 2021 at 13:26
• Yes, thank you! I would love a result giving sufficient conditions for $D\setminus S$ to be shy. But I'd also love results saying $S$ is "most of" $D$ in other senses---a measure theoretic sense was just one example. Commented Mar 30, 2021 at 18:56
• Sorry, I have a correction to make to my previous comment: instead it should have said that a set is called prevalent if its complement is shy. Commented Mar 30, 2021 at 19:08