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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
Published electronically: May 18, 2000
MathSciNet review: 1690986
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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 $f^{*}\vee g^{*}$.

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Additional Information

S. P. Fitzpatrick
Affiliation: Department of Mathematics and Statistics, University of Western Australia, Nedlands 6907, Australia

S. Simons
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106-3080

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