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Dual characterizations of relative continuity of convex functions

Part of: Real functions Nonlinear operators and their properties General convexity

Published online by Cambridge University Press:  09 April 2009

J. Benoist  and
A. Daniilidis
J. Benoist
Affiliation:
LACO, CNRS UPRES 6090 Faculté des Sciences Université de Limoges123, avenue Albert Thomas 87060 Limoges, CedexFrance e-mail: joel.benoist@unilim.fr
A. Daniilidis
Affiliation:
CNRS ERS 2055 Laboratoire de Mathématiques Appliquées Université de Pau et des Pays de l'Adour avenue de l's Université 64000 PauFrance e-mail: aris.daniilidis@univ-pau.fr
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Abstract

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Various properties of continuity for the class of lower semicontinuous convex functions are considered and dual characterizations are established. In particular, it is shown that the restriction of a lower semicontinuous convex function to its domain (respectively, domain of subdifferentiability) is continuous if and only if its subdifferential is strongly cyclically monotone (respectively, σ-cyclically monotone).

Keywords

convexitycontinuitysubdifferentialcyclic monotonicity.

MSC classification

Secondary: 47H05: Monotone operators and generalizations 52A41: Convex functions and convex programs 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 2 , April 2001 , pp. 211 - 224
Copyright
Copyright © Australian Mathematical Society 2001

References

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