Convexity is important in theoretical aspects of mathematics and
also for economists and physicists. In this monograph the author
provides a comprehensive insight into convex sets and functions
including the infinite-dimensional case and emphasizing the
analytic point of view. Chapter one introduces the reader to the
basic definitions and ideas that play central roles throughout the
book. The rest of the book is divided into four parts: convexity
and topology on infinite-dimensional spaces; Loewner's theorem;
extreme points of convex sets and related issues, including the
Krein-Milman theorem and Choquet theory; and a discussion of
convexity and inequalities. The connections between disparate
topics are clearly explained, giving the reader a thorough
understanding of how convexity is useful as an analytic tool. A
final chapter overviews the subject's history and explores further
some of the themes mentioned earlier. This is an excellent resource
for anyone interested in this central topic.
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