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Measures of symmetry for convex sets and stability

Record Type:	Electronic resources : Monograph/item
Title/Author:	Measures of symmetry for convex sets and stabilityby Gabor Toth.
Author:	Toth, Gabor.
Published:	Cham :Springer International Publishing :2015.
Description:	xii, 278 p. :ill. (some col.), digital ;24 cm.
Contained By:	Springer eBooks
Subject:	Mathematics.
Online resource:	http://dx.doi.org/10.1007/978-3-319-23733-6
ISBN:	9783319237336$q(electronic bk.)

Measures of symmetry for convex sets and stability
Toth, Gabor.

Measures of symmetry for convex sets and stability[electronic resource] /by Gabor Toth. - Cham :Springer International Publishing :2015. - xii, 278 p. :ill. (some col.), digital ;24 cm. - Universitext,0172-5939. - Universitext..

First Things First on Convex Sets -- Affine Diameters and the Critical Set -- Measures of Stability and Symmetry -- Mean Minkowski Measures.

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 Grunbaum'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 Caratheodory, 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 Grunbaum'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.

ISBN: 9783319237336$q(electronic bk.)

Standard No.: 10.1007/978-3-319-23733-6doiSubjects--Topical Terms:

184409
Mathematics.

LC Class. No.: QA639.5

Dewey Class. No.: 516.1

Measures of symmetry for convex sets and stability
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520 $a 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 Grunbaum'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 Caratheodory, 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 Grunbaum'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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