Measures of Symmetry for Convex Sets and Stability / Universitext (PDF)
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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 thebook's subject.
- Autor: Gabor Toth
- 2015, 1st ed. 2015, 278 Seiten, Englisch
- Verlag: Springer-Verlag GmbH
- ISBN-10: 3319237330
- ISBN-13: 9783319237336
- Erscheinungsdatum: 26.11.2015
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- Dateiformat: PDF
- Größe: 4.86 MB
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“This is a graduate-level textbook on convex geometry in finite-dimensional Euclidean spaces, which has some interesting special features. … Each chapter has illustrating figures and concludes with exercises … . The book has a surprising appendix, where certain of the symmetry measures are applied to convex bodies … . This book is an unconventional introduction to convexity, full of appealing intuitive geometry; it may equally well serve the beginner and the experienced researcher in the field.” (Rolf Schneider, zbMATH 1335.52002, 2016)
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