In the second edition of Geometric Methods and Applications, Jean Gallier expanded some issues and added quite a few topics (for the first edition see Zbl 1031.53001):

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The chapter on convex sets is extended by Carathéodory’s theorem for convex cones and Radon’s theorem for pointed cones. The introduction of centerpoints is also new.

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Separating and supporting hyperplanes are introduced, and the separation of open or closed convex sets by hyperplanes is discussed. Farkas’s lemma and its application to linear programming is now presented. Moreover, the existence of supporting hyperplanes for boundary points of closed convex sets is proved.

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Principal component analysis (PCA) and its relation to singular value decomposition (SVD) is extensively discussed.

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Quadratic optimization problems (also with linear, affine or quadratic constraints) are treated much more extensively than in the first edition. The task of contour grouping stemming from the field of computer vision and its connection with quadratic optimization is dealt with.

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Schur complements and their connection with symmetric positive semidefinite matrices are explained.

Summarizing, one can say that the book contains a valuable collection of modern geometric methods and algorithms readily prepared for solving problems occurring in computer science and engineering.
In the last decade the first edition of Gallier’s monograph has turned out to be a benchmark in its field. The second edition is even more comprehensive and puts more emphasis on the links between different fields. It can be recommended to anybody who is interested in modern geometry and its applications.