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Journal of Convex Analysis 17 (2010), No. 3&4, 915--924
Copyright Heldermann Verlag 2010

Fréchet-Legendre Functions and Reflexive Banach Spaces

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

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

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 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.

Keywords: Convex Function, Legendre function, essentially smooth, essentially strictly convex, Frechet differentiability, Fenchel duality.

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

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