January  2008, 7(1): 119-124. doi: 10.3934/cpaa.2008.7.119

Refinement of the Benoist theorem on the size of Dini subdifferentials

1. 

Université de Nice-Sophia Antipolis, Laboratoire J.A. Dieudonné, Parc Valrose, 06108 Nice Cedex 02, France

Received  September 2006 Revised  May 2007 Published  October 2007

Given a lower semicontinuous function $f:\mathbb R^n \rightarrow \mathbb R \cup$ {$+\infty$}, we prove that the set of points of $\mathbb R^n$ 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.
Citation: Ludovic Rifford. Refinement of the Benoist theorem on the size of Dini subdifferentials. Communications on Pure & Applied Analysis, 2008, 7 (1) : 119-124. doi: 10.3934/cpaa.2008.7.119
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