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

 

Minimal convex functions bounded below by the duality product


Authors: J.-E. Martínez-Legaz and B. F. Svaiter
Journal: Proc. Amer. Math. Soc. 136 (2008), 873-878
MSC (2000): Primary 47H05; Secondary 52A41, 26B25
Published electronically: November 30, 2007
MathSciNet review: 2361859
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Abstract: It is well known that the Fitzpatrick function of a maximal monotone operator is minimal in the class of convex functions bounded below by the duality product. Our main result establishes that, in the setting of reflexive Banach spaces, the converse also holds; that is, every such minimal function is the Fitzpatrick function of some maximal monotone operator. Whether this converse also holds in a nonreflexive Banach space remains an open problem.


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Additional Information

J.-E. Martínez-Legaz
Affiliation: Departament d’Economia i d’Història Econòmica, Universitat Autònoma de Barce- lona, 08193 Bellaterra, Spain
Email: JuanEnrique.Martinez@uab.es

B. F. Svaiter
Affiliation: Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorinha 110, Jardim Botânico, Rio de Janeiro, CEP 22460-320, Brazil
Email: benar@impa.br

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09176-9
PII: S 0002-9939(07)09176-9
Received by editor(s): June 20, 2006
Published electronically: November 30, 2007
Additional Notes: The first author was partially supported by the Ministerio de Ciencia y Tecnología, Project MTM2005-08572-C03-03. He also thanks the support of the Barcelona Economics Program of CREA
The second author was partially suported by CNPq grant n.\ 300755/2005-8 and Edital Universal 476842/03-2
This work was initiated during a visit of the second author to the Universitat Autònoma de Barcelona in March 2006
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2007 American Mathematical Society