Measures Of Symmetry For Convex Sets And Stability
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Measures of Symmetry for Convex Sets and Stability
This textbook treats two important and related matters in convex geometry: the quantification of symmetry of a convex set--measures of symmetry--and the degree to which convex sets that nearly minimize such measures of symmetry are themselves nearly symmetric--the phenomenon of stability. By gathering the subject's core ideas and highlights around Grünbaum's general notion of measure of symmetry, it paints a coherent picture of the subject, and guides the reader from the basics to the state-of-the-art. The exposition takes various paths to results in order to develop the reader's grasp of the unity of ideas, while interspersed remarks enrich the material with a behind-the-scenes view of corollaries and logical connections, alternative proofs, and allied results from the literature. Numerous illustrations elucidate definitions and key constructions, and over 70 exercises--with hints and references for the more difficult ones--test and sharpen the reader's comprehension. The presentation includes: a basic course covering foundational notions in convex geometry, the three pillars of the combinatorial theory (the theorems of Carathéodory, Radon, and Helly), critical sets and Minkowski measure, the Minkowski-Radon inequality, and, to illustrate the general theory, a study of convex bodies of constant width; two proofs of F. John's ellipsoid theorem; a treatment of the stability of Minkowski measure, the Banach-Mazur metric, and Groemer's stability estimate for the Brunn-Minkowski inequality; important specializations of Grünbaum's abstract measure of symmetry, such as Winternitz measure, the Rogers-Shepard volume ratio, and Guo's Lp -Minkowski measure; a construction by the author of a new sequence of measures of symmetry, the kth mean Minkowski measure; and lastly, an intriguing application to the moduli space of certain distinguished maps from a Riemannian homogeneous space to spheres--illustrating the broad mathematical relevance of the book's subject.
Measures of Symmetry for Convex Sets and Stability
This textbook treats two important and related matters in convex geometry: the quantification of symmetry of a convex set—measures of symmetry—and the degree to which convex sets that nearly minimize such measures of symmetry are themselves nearly symmetric—the phenomenon of stability. By gathering the subject’s core ideas and highlights around Grünbaum’s general notion of measure of symmetry, it paints a coherent picture of the subject, and guides the reader from the basics to the state-of-the-art. The exposition takes various paths to results in order to develop the reader’s grasp of the unity of ideas, while interspersed remarks enrich the material with a behind-the-scenes view of corollaries and logical connections, alternative proofs, and allied results from the literature. Numerous illustrations elucidate definitions and key constructions, and over 70 exercises—with hints and references for the more difficult ones—test and sharpen the reader’s comprehension. The presentation includes: a basic course covering foundational notions in convex geometry, the three pillars of the combinatorial theory (the theorems of Carathéodory, Radon, and Helly), critical sets and Minkowski measure, the Minkowski–Radon inequality, and, to illustrate the general theory, a study of convex bodies of constant width; two proofs of F. John’s ellipsoid theorem; a treatment of the stability of Minkowski measure, the Banach–Mazur metric, and Groemer’s stability estimate for the Brunn–Minkowski inequality; important specializations of Grünbaum’s abstract measure of symmetry, such as Winternitz measure, the Rogers–Shepard volume ratio, and Guo’s Lp -Minkowski measure; a construction by the author of a new sequence of measures of symmetry, the kth mean Minkowski measure; and lastly, an intriguing application to the moduli space of certain distinguished maps from a Riemannian homogeneous space to spheres—illustrating the broad mathematical relevance of the book’s subject.
Convexity from the Geometric Point of View: Exercises and Solutions
This book provides the solutions to all 347 exercises contained in the text Convexity from the Geometric Point of View, published in the same Cornerstones series. All these exercises are restated and numbered analogously to those in the original text. The corresponding solutions follow each exercise. Besides the discussion of all solutions, some additional facts about the main text are sprinkled throughout. Sections of further reading are posted to the ends of each chapter supplying the reader with background literature to selected notions and tools that play a role in the exercises and/or solutions to the chapter. The original text gives a comprehensive introduction to the “common core” of convex geometry and is suitable as a primary text for courses in convex geometry and in discrete geometry (including polytopes). Additionally, it can be used as a single reference for a complete introduction to convex geometry. The content coverage is sufficiently broad that the reader may gain a glimpse of the entire breadth of the field, various subfields, and interesting connections to neighboring disciplines. Mainly directed to graduate and advanced undergraduates, the original text is self-contained in such a way that it can be read by anyone who has standard undergraduate knowledge of analysis and of linear algebra. The same is true for this book of solutions.