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 subjects core ideas and highlights around Grnbaums 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 readers 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 readers comprehension. The presentation includes: a basic course covering foundational notions in convex geometry, the three pillars of the combinatorial theory (the theorems of Carathodory, 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. Johns ellipsoid theorem; a treatment of the stability of Minkowski measure, the BanachMazur metric, and Groemers stability estimate for the BrunnMinkowski inequality; important specializations of Grnbaums abstract measure of symmetry, such as Winternitz measure, the RogersShepard volume ratio, and Guos 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 spheresillustrating the broad mathematical relevance of thebooks subject.