Refinement of the Benoist theorem on the size of Dini subdifferentials

Ludovic Rifford

doi:10.3934/cpaa.2008.7.119

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2008, Volume 7, Issue 1: 119-124. Doi: 10.3934/cpaa.2008.7.119

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Refinement of the Benoist theorem on the size of Dini subdifferentials

Ludovic Rifford^1,

1.
Université de Nice-Sophia Antipolis, Laboratoire J.A. Dieudonné, Parc Valrose, 06108 Nice Cedex 02

Received: August 31, 2006

Revised: April 30, 2007

Published: December 2007

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Abstract

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.

Keywords:

Nonsmooth analysis,
generalized differentials.

Mathematics Subject Classification: Primary: 49J52, 26B35; Secondary: 26BXX.

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