Abstract

H. H. Bauschke, J. M. Borwein and P. L. Combettes ["Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces", Communications in Contemp. Mathematics 3 (2001) 615–647] showed how to extend naturally the classical definitions of essential smoothness and essential strict convexity from functions on Rn\mathbb{R}^n in a compatible fashion to any Banach space. They were able, among other things, to show that substantial duality results hold for Legendre functions in reflexive spaces. That article focused on essential smoothness in the Gâteaux sense. Our goal herein is to show that similar results hold for Fréchet smoothness and to study related properties of such functions on reflexive Banach spaces.

Author information

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

Jonathan M. Borwein

Centre for Computer Assisted Research Mathematics and its Applications, University of Newcastle, Callaghan NSW 2308, Australia

jborwein@newcastle.edu.au

Jon Vanderwerff

Department of Mathematics, La Sierra University, Riverside, CA 92515, U.S.A.

jvanderw@lasierra.edu

Keywords

  • Convex Function
  • Legendre function
  • essentially smooth
  • essentially strictly convex
  • Fréchet differentiability
  • Fenchel duality

Mathematics Subject Classification

52A41; 46G05, 46N10, 49J50, 90C25

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Suggested citation

J. M. Borwein, J. Vanderwerff. “Fréchet-Legendre Functions and Reflexive Banach Spaces.” Journal of Convex Analysis 17 (2010), No. 3&4, 915–924. https://doi.org/10.68381/jca17057

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.