Convex functions : constructions, characterizations and counterexamples /

The best notions of mathematics (think of finiteness, continuity, linearity, connectedness, commutativity, exactness, smoothness, etc.) manage both to arise across broad ranges of context, yet nevertheless carry strong and subtle implications. Characteristically, such notions form the cornerstones for whole disciplines, and indeed, in terms of convexity, according to Borwein (Univ. of Newcastle, Australia) and Vanderwerff (La Sierra Univ.), three distinct disciplines at least. Directing readers to Convex Analysis by R. T. Rockafellar (1970) and Convex Optimization by S. P. Boyd and L. Vandenberghe (2004) for two other main branches of convexity, the authors here concentrate on those properties of convex functions that reflect the structure of the normed spaces where the functions have their definition. This masterful book emerges immediately as the de facto canonical source on it subject, and thus as a vital reference for students of Banach space geometry, functional analysis, analytic inequalities, and needless to say, any aspect of convexity. In the exercises and asides, which maintain lively rapport with a spectrum of mathematical concerns, one finds mention of unexpected topics such as the brachistochrone problem and the Riemann zeta function. Truly then, anyone interested in nearly any branch of mathematical analysis should at least browse this book. Summing Up: Essential. Upper-division undergraduates and above. D. V. Feldman University of New Hampshire