Measures of symmetry for convex sets and stability [electronic resource] / by Gabor Toth.
Toth, Gabor. author.
2015
QA639.5
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Title Measures of symmetry for convex sets and stability [electronic resource] / by Gabor Toth.
Author Toth, Gabor. author.
ISBN 9783319237336 electronic book
3319237330 electronic book
9783319237329
3319237322
DOI https://doi.org/10.1007/978-3-319-23733-6
Published Cham : Springer, 2015.
Language English
Description 1 online resource (xii, 278 pages) : illustrations.
Item Number 10.1007/978-3-319-23733-6 doi
Call Number QA639.5
Dewey Decimal Classification 516.1
Summary 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.
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Digital File Characteristics text file PDF
Series Universitext.
Available in Other Form Print version: 9783319237329
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Record Appears in Online Resources > Ebooks
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First Things First on Convex Sets
Affine Diameters and the Critical Set
Measures of Stability and Symmetry
Mean Minkowski Measures.

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