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.

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    $\begingroup$ 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. $\endgroup$
    – Leo Moos
    Mar 30, 2021 at 13:26
  • $\begingroup$ 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. $\endgroup$ Mar 30, 2021 at 18:56
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    $\begingroup$ 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. $\endgroup$
    – Leo Moos
    Mar 30, 2021 at 19:08


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