Refinement of the Benoist theorem on the size of Dini subdifferentials

Ludovic Rifford (1)

1 JAD - Laboratoire Jean Alexandre Dieudonné

Given a lower semicontinuous function $f : \R^n \rightarrow \R \cup \{+\infty\}$, we prove that the set of points of $\R^n$ where the lower Dini subdifferential has convex dimension $k$ is countably $(n-k)$-rectifiable. In this way, we extend a theorem of Benoist(see [1, Theorem 3.3]), and as a corollary we obtain a classical result concerning the singular set of locally semiconcave functions.

Domains: General Mathematics [math.GM]; Optimization and Control [math.OC]

Ludovic Rifford. Refinement of the Benoist theorem on the size of Dini subdifferentials. Communications on Pure and Applied Mathematics, 2008, 7 (1), pp.119-124. ⟨hal-00769097⟩