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Abstract Convex Analysis (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts) ReviewSinger's book studies abstract convexity, and it will appear to mathematicians with an interest in approximation theory, general topology, lattice theory, or convex analysis---especially those interested in generalizing duality theory from real vector spaces to convexity systems, etc. I'll register one complaint: I cannot find the definition of "complete lattice" in the book! (I hope Birkhoff's definition is intended.)Singer is a gifted expositor who has previously written books on Bases in Banach Spaces and Approximation Theory, which often exhibit virtuosity in "hard analysis". In contrast, this book is quite abstract, although a nice introduction reviews convex duality and quasi-convex ("surrogate") duality as motivation.
A related book is A. M. Rubinov's Abstract Convexity and Optimization (the title may be slightly wrong), which has been published by Kluwer in its Nonconvex Optimization Series. Rubinov's book would make collateral reading for researchers in global minimization algorithms.
I wrote this review to warn that the previous reviewer must have been writing about another book, apparently in signal processing!Abstract Convex Analysis (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts) Overview
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