Convex Functions: Constructions, Characterizations and Counterexamples

Abstract

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.

Monotone Operators Without Enlargements

“…We adopt the notation used in the books [15,Chapter 2] and [12,31,32]. Given a subset C of X, int C is the interior of C, C is the norm closure of C. The support function of C, written as σ C , is defined by σ C (x * ) := sup c∈C c, x * .…”

“…The name is motivated by the celebrated theorem of Brønsted and Rockafellar [32,15] which can be stated now as saying that all closed convex subgradients are of type (BR).…”

“…Their study in the context of Banach spaces, and in particular nonreflexive ones, arises naturally in the theory of partial differential equations, equilibrium problems, and variational inequalities. For a detailed study of these operators, see, e.g., [12,13,14], or the books [6,15,19,25,31,32,30,41,42].…”

Right Bregman nonexpansive operators in Banach spaces

“…If X is a uniformly smooth and uniformly convex Banach space and f = (1/2) · 2 , then ∇f = J, where J is the duality mapping of the space X. Hence inequality (12) in Definition 2.3 (i*) becomes…”

“…Its principal branch on the real axis is concave and increasing, and its domain is (−1/e, +∞) (cf. [10,12]). Then T (x) = W (x) is R-BFNE with respect to BS and e x on (0, +∞) and with respect to FD on (0, 1).…”