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Book review

Author(s) Udriste, Constantin
Title Convex Functions and Optimization Methods on Riemannian Manifolds
Publisher Kluwer Academic Publishers
Year of publication 1994
Reviewed by Mihai Postolache

This book, written by an expert in the field, comprises advanced research concerning the convex functions and the optimization methods on Riemannian manifolds, the suited mathematical frame which allows to debate about convexity and which leads to a deeply understanding of this important concept. Based on original researches of this famous Romanian scientist, the book is written with nice harmony and charm and is thought as a monograph including advanced topics in Differential Geometry, Analysis on Manifolds, Mathematical Programming and Optimization. Also this book gives an interdisciplinary perspective with the dynamical systems numerical analysis, and numerical simulation using Turbo Pascal and Turbo C programming.

The purpose of the book is to present a systematic exposition of the Riemannian convexity with applications in Mathematical Modelling, Industrial Mathematics, Dynamical Systems, and Theoretical Physics. The first part of the book deals with metric properties of Riemannian manifolds, the first and second variation of the p-energy of a curve. The second part refers to convex functions on Riemannian manifolds and some examples, flows, convexity and energies, semidefinite Hessians and applications. The third part is devoted to the theory of minimization of functions on Riemannian manifolds. Here we can find new ideas about the numerical approximation of geodesics and the Riemannian description of the descent algorithms. The fourth part is devoted to some appendices where special attention is paid to the completeness and convexity on Finsler spaces. The material is organized so as to introduce as soon as possible these concepts. A comprehensive bibliography is included, too.

The style is pedagogically ideal in proceeding from general to special or particular and also there are presented some illustrative examples and open problems. The text itself is clear. This is a good book for those who want to get a good introduction to convex functions and to optimization methods on Riemannian manifolds. I recommend this book to anybody interested in these topics: mathematicians, theoretical physicists as well as to anybody interested in geometrical modelling. Also, parts of this monograph can be used as a course for undergraduate students in mathematics or for graduate students in applied sciences. As prerequisites for using the book, there is a good foundation in the theory of differential geometry and a general background in variational calculus and mathematical modelling. Applied scientists will be able to find concise yet detailed descriptions of new mathematical ideas that "must be written down somewhere", but rarely are.