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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The size of the Dini subdifferential


Author: Joël Benoist
Journal: Proc. Amer. Math. Soc. 129 (2001), 525-530
MSC (2000): Primary 26A16, 26A24
Published electronically: September 18, 2000
MathSciNet review: 1707505
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Abstract:

Given a lower semicontinuous function $f:\mathbb{R}^h \rightarrow \mathbb{R} \cup \{+\infty\}$, we prove that the points of $\mathbb{R}^h$, where the lower Dini subdifferential contains more than one element, lie in a countable union of sets which are isomorphic to graphs of some Lipschitzian functions defined on $\mathbb{R}^{h-1}$. Consequently, the set of all these points has a null Lebesgue measure.


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Additional Information

Joël Benoist
Affiliation: Maître de Conférences, LACO, CNRS-ESA 6090, Université de Limoges, 87 060 Limoges, France
Email: benoist@unilim.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05549-0
PII: S 0002-9939(00)05549-0
Keywords: Lower semicontinuous function, Dini subdifferential, proximal subdifferential, countability, null Lebesgue measure set
Received by editor(s): October 7, 1998
Received by editor(s) in revised form: May 3, 1999
Published electronically: September 18, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society