I think the best way to begin is to quote some introductory remarks from the book. Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characterizations, treating convex functions in both Euclidean and Banach spaces.. .. The book can either be read sequentially as a graduate text, or dipped into by researchers and practitioners. Each chapter contains a variety of concrete examples and over 600 exercises are included, ranging in difficulty from early graduate to research level.

Abstract. Given a Banach space (X, · ), we study the connection between uniformly convex functions f : X → R bounded above by · p and the existence of norms on X with moduli of convexity of power type. In particular, we show that there exists a uniformly convex function f : X → R bounded above by · 2 if and only if X admits an equivalent norm with modulus of convexity of power type 2.

In this paper we study approximations of convex functions by twice Gâteaux differentiate convex functions. We prove that convex functions (respectively norms) can be approximated by twice Gâteaux differentiate convex functions (respectively norms) in separable Banach spaces which have the Radon-Nikody m property and admit twice Gâteaux differentiable bump functions. New characterizations of spaces isomorphic to Hilbert spaces are shown. Locally uniformly rotund norms that are limits of Ck-smooth norms are constructed in separable spaces which admit Ck-smooth norms.

We study some structural properties of real linear spaces equipped with norms that may take the value infinity but that otherwise satisfy the properties of conventional norms. A description is given of the finest locally convex topology weaker than the extended norm topology for which addition and scalar multiplication are jointly continuous. We also study bornologies and provide a characterization of relatively weakly compact sets in these spaces. It is shown that complemented and projection complemented closed subspaces can be different in extended Banach spaces. Particular attention is given to extended normed spaces whose subspace of vectors of finite norm has finite codimension.Mathematics Subject Classification (2010) 46B20 · 46B28 · 46A17 · 46A55 a notre ami Lionel Thibault J. Vanderwerff

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