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 setmeasures of symmetryand the degree to which convex sets that nearly minimize such measures of symmetry are themselves nearly symmetricthe 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 exerciseswith hints and references for the more difficult onestest 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 MinkowskiRadon 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 BanachMazur metric, and Groemer's stability estimate for the BrunnMinkowski inequality; important specializations of Grünbaum's abstract measure of symmetry, such as Winternitz measure, the RogersShepard volume ratio, and Guo's Lp -Minkowski measure; a construction by the author of a new sequence of measures of symmetry, the k th mean Minkowski measure; and lastly, an intriguing application to the moduli space of certain distinguished maps from a Riemannian homogeneous space to spheresillustrating the broad mathematical relevance of thebook's subject.

Starting from the basics, provides a coherent and up-to-date account of various quantitative measures of symmetry arising in convex geometry Contains over 90 challenging problems, with hints and references to aid the reader Introduces a new sequence of mean Minkowski measures associated to a convex body

Autorentext
Gabor Toth is Chair of the Department of Mathematical Sciences at Rutgers University, Camden. His research interests include harmonic maps and minimal immersions and convex geometry. He is the author of Glimpses of Algebra and Geometry, as well as Finite Möbius Groups, Spherical Minimal Immersions, and Moduli.

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