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 | References | Similar Articles | Additional Information

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- functions. In finite dimensions the class of primal lower-nice functions also contains the composition of a convex function with a 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