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Convex composite functions in Banach spaces
and the primal lower-nice property


Authors: C. Combari, A. Elhilali Alaoui, A. Levy, R. Poliquin and L. Thibault
Journal: Proc. Amer. Math. Soc. 126 (1998), 3701-3708
MSC (1991): Primary 58C20; Secondary 49J52
DOI: https://doi.org/10.1090/S0002-9939-98-04324-X
MathSciNet review: 1451793
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Abstract: Primal lower-nice functions defined on Hilbert spaces provide examples of functions that are ``integrable'' (i.e. of functions that are determined up to an additive constant by their subgradients). The class of primal lower-nice functions contains all convex and lower-$C^2$ functions. In finite dimensions the class of primal lower-nice functions also contains the composition of a convex function with a $C^2$ mapping under a constraint qualification. In Banach spaces certain convex composite functions were known to be primal lower-nice (e.g. a convex function had to be continuous relative to its domain). In this paper we weaken the assumptions and provide new examples of convex composite functions defined on a Banach space with the primal lower-nice property. One consequence of our results is the identification of new examples of integrable functions on Hilbert spaces.


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Additional Information

C. Combari
Affiliation: Université de Montpellier II, Laboratoire Analyse Convexe, Place Eugene Bataillon, 34095 Montpellier Cedex 5, France

A. Elhilali Alaoui
Affiliation: Université de Montpellier II, Laboratoire Analyse Convexe, Place Eugene Bataillon, 34095 Montpellier Cedex 5, France
Address at time of publication: Falculté des Sciences et Techniques de Marrakech, Université Cadi Ayad, B.P. 618, Marrakech, Maroc

A. Levy
Affiliation: Department of Mathematics, Bowdoin College, Brunswick, Maine 04011
Email: alevy@bowdoin.edu

R. Poliquin
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: rene.poliquin@ualberta.ca

L. Thibault
Affiliation: Université de Montpellier II, Laboratoire Analyse Convexe, Place Eugene Bataillon, 34095 Montpellier Cedex 5, France

DOI: https://doi.org/10.1090/S0002-9939-98-04324-X
Keywords: Primal lower-nice functions, subdifferential, convex composite functions, integrable functions
Received by editor(s): February 16, 1996
Received by editor(s) in revised form: November 27, 1996
Additional Notes: The research of R. Poliquin was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983.
Communicated by: Dale Alspach
Article copyright: © Copyright 1998 American Mathematical Society

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