Fréchet-Legendre Functions and Reflexive Banach Spaces

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

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.