Abstract

Let (X,∥⋅∥)(X, \|\cdot\|) be a Banach space and f ⁣:X→R∪{∞}f\colon X \to \mathbb{R} \cup \{\infty\} be a proper function. Then the Fenchel conjugate of ff is the function f∗ ⁣:X∗→R∪{∞}f^*\colon X^* \to \mathbb{R} \cup \{\infty\} defined by, f∗(x∗):=sup⁡{(x∗−f)(x):x∈X}.f^*(x^*):= \sup\{(x^*-f)(x):x \in X\}. In this article we will prove a theorem more general than the following. Theorem: Let f ⁣:X→R∪{∞}f\colon X \to \mathbb{R} \cup \{\infty\} be a proper function on a Banach space (X,∥⋅∥)(X,\|\cdot\|). If there is a nonempty open subset AA of Dom(f∗)\mathrm{Dom}(f^*) such that argmax(x∗−f)≠∅\mathrm{argmax}(x^*-f) \not= \varnothing for each x∗∈Ax^* \in A, then there is a dense and GδG_\delta subset RR of AA such that (x∗−f) ⁣:X→R∪{−∞}(x^*-f) \colon X \to \mathbb{R} \cup \{-\infty\} has a strong maximum for each x∗∈Rx^* \in R. In addition, if 0∈A0 \in A and 0<ε0<\varepsilon then there is an x∗∈X∗x^* \in X^* with ∥x∗∥<ε\|x^*\| < \varepsilon such that (x∗−f) ⁣:X→R∪{−∞}(x^* -f) \colon X \to \mathbb{R} \cup \{-\infty\} has a strong maximum.

Author information

Contact details are reproduced from the original publication and may be historical.

Warren B. Moors

Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand

w.moors@auckland.ac.nz

Neşet Özkan Tan

Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand

neset.tan@auckland.ac.nz

Keywords

  • Variational theorem
  • James' weak compactness theorem
  • convex analysis

Mathematics Subject Classification

46B20; 46B10, 46B50

References

  1. J. M. Borwein, W. B. Moors, X. Wang: Generalized subdifferentials: a Baire categorical approach, Trans. Amer. Math. Soc. 353(10) (2001) 3875–3893.https://doi.org/10.1090/S0002-9947-01-02820-3Google Scholar
  2. J. P. R. Christensen: Theorems of Namioka and R. E. Johnson type for upper semicontinuous and compact valued set-valued mappings, Proc. Amer. Math. Soc. 86(4) (1982) 649–655.https://doi.org/10.1090/S0002-9939-1982-0674099-0Google Scholar
  3. N. Dunford, J. T. Schwartz: Linear Operators. I: General Theory, Interscience Publishers, New York et al. (1958).Google Scholar
  4. M. J. Fabian: Gâteaux Differentiability of Convex Functions and Topology, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley and Sons, New York (1997).Google Scholar
  5. J. R. Giles, P. S. Kenderov, W. B. Moors, S. D. Sciffer: Generic differentiability of convex functions on the dual of a Banach space, Pacific J. Math. 172(2) (1996) 413–431.https://doi.org/10.2140/pjm.1996.172.413Google Scholar
  6. A. Grothendieck: Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952) 168–186.https://doi.org/10.2307/2372076Google Scholar
  7. R. C. James: Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964) 129–140.https://doi.org/10.1090/S0002-9947-1964-0165344-2Google Scholar
  8. L. Jokl: Minimal convex-valued weak* USCO correspondences and the Radon-Nikodým property, Comment. Math. Univ. Carolin. 28(2) (1987) 353–376.Google Scholar
  9. M. Krein: Sur quelques questions de la géométrie des ensembles convexes situés dans un espace linéaire normé et complet, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 14 (1937) 5–7.Google Scholar
  10. W. B. Moors: A gentle introduction to James’ weak compactness theorem and beyond, Methods Funct. Anal. Topology 25(1) (2019) 35–83.Google Scholar
  11. W. B. Moors, S. J. White: An elementary proof of James’ characterisation of weak compactness II, Bull. Aust. Math. Soc. 95(1) (2017) 133–137.https://doi.org/10.1017/S0004972716000599Google Scholar
  12. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics 1364, Springer, Berlin (1993).Google Scholar
  13. R. S. Phillips: On weakly compact subsets of a Banach space, Amer. J. Math. 65 (1943) 108–136.https://doi.org/10.2307/2371776Google Scholar

Suggested citation

W. B. Moors, N. O. Tan. “An Abstract Variational Theorem.” Journal of Convex Analysis 26 (2019), No. 4, 1125–1144. https://doi.org/10.68381/jca26062

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.