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On the pointwise maximum of convex functions


Authors: S. P. Fitzpatrick and S. Simons
Journal: Proc. Amer. Math. Soc. 128 (2000), 3553-3561
MSC (2000): Primary 46N10, 49J52, 49N15
DOI: https://doi.org/10.1090/S0002-9939-00-05449-6
Published electronically: May 18, 2000
MathSciNet review: 1690986
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the conjugate of the maximum, $f \vee g$, of $f$ and $g$ when $f$ and $g$ are proper convex lower semicontinuous functions on a Banach space $E$. We show that $(f \vee g)^{**} = f^{**} \vee g^{**}$ on the bidual, $E^{**}$, of $E$ provided that $f$and $g$ satisfy the Attouch-Brézis constraint qualification, and we also derive formulae for $(f \vee g)^{*}$ and for the ``preconjugate'' of $f^{*}\vee g^{*}$.


References [Enhancements On Off] (What's this?)

  • [1] H. Attouch and H. Brézis, Duality for the sum of convex funtions in general Banach spaces, Aspects of Mathematics and its Applications, J. A. Barroso, ed, Elsevier Science Publishers, 1986, pp. 125-133. MR 87m:90095
  • [2] K. Fan, Minimax theorems, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 42-47. MR 14:1109f
  • [3] J.-P. Gossez, Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs, J. Math. Anal. Appl. 34 (1971), 371-395. MR 47:2442
  • [4] H. König, Über das Von Neumannsche Minimax-Theorem, Arch. Math. 19 (1968), 482-487. MR 39:1947
  • [5] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pac. J. Math. 33 (1970), 209-216. MR 41:7432
  • [6] S. Simons, Critères de faible compacité en termes du théorème de minimax, Seminaire Choquet 1970/1971, no. 24, 5 pages. MR 57:17218
  • [7] S. Traoré and M. Volle, On the level sum of two convex functions on Banach spaces, J. Convex Analysis 3 (1996), 141-151. MR 97m:46116
  • [8] M. Volle, Sous-differential d'une enveloppe supérieure de fonctions convexes, C. R. Acad. Sci. Paris 317 (1993), 845-849. MR 94h:49035

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Additional Information

S. P. Fitzpatrick
Affiliation: Department of Mathematics and Statistics, University of Western Australia, Nedlands 6907, Australia
Email: fitzpatr@maths.uwa.edu.au

S. Simons
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106-3080
Email: simons@math.ucsb.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05449-6
Keywords: Banach space, convex function, conjugate, biconjugate, maximum, Attouch-Br\'{e}zis constraint qualification, preconjugate
Received by editor(s): May 11, 1998
Received by editor(s) in revised form: January 29, 1999
Published electronically: May 18, 2000
Dedicated: This paper is dedicated to Professor Robert Phelps
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

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