Measures of symmetry for convex sets and stability
This is a graduate-level textbook on convex geometry in finite-dimensional Euclidean spaces, which has some interesting special features. Using the author's words, it starts from the beginning, guides the reader through the necessary background, and ends up in recent research. A special feature is that the main part of the book is developed along two particular, though comprehensive, topics of classical convex geometry, namely measures of symmetry, and stability of geometric inequalities. A measure of symmetry is a continuous real function on the space of convex bodies, perhaps with specific additional properties, that attains one of its extrema precisely at symmetric bodies, so that this function generally measures the deviation from symmetric shape. A stability result for a geometric inequality for convex bodies estimates how much a convex body deviates from an extremal body if it satisfies the inequality with approximate equality. The well-known survey articles by \textit{Branko Grünbaum} on ``Measures of symmetry for convex sets'' [Proc. Sympos. Pure Math. 7, 233--270 (1963; Zbl 0142.20503)] and by \textit{Helmut Groemer} on ``Stability of geometric inequalities'' [in: Handbook of convex geometry. Volume A. Amsterdam: North-Holland. 125--150 (1993; Zbl 0789.52001)] showed impressively how rich these topics were already decades ago. The first chapter of this book lays the necessary foundations on convex sets in finite dimensions and treats, for example, the Hausdorff distance (and other distances), the Blaschke selection theorem, separation theorems, Minkowski's theorem on extreme points, mixed volumes (briefly), the combinatorial theorems of Carathéodory, Radon and Helly, and classical results on circumradius and inradius. The chapter concludes with a detailed proof, using Painlevé-Kuratowski convergence, of a Helly-type theorem of Klee on unbounded convex sets. The second chapter treats various notions of immediate geometric appeal, but also preparing the study of symmetry measures. Here we find the distortion ratio, affine diameters, critical sets, a detailed proof of Klee's inequality for the Minkowski measure of symmetry and the dimension of the critical set, and some essentials on bodies of constant width. The third chapter then presents various major results on symmetry measures and stability, on the way also dealing with important tools, such as Banach-Mazur distance and John's ellipsoid theorem. Treated in detail are the Minkowski measure of symmetry and its variants, corresponding stability results, Groemer's stability estimate for the Brunn-Minkowski inequality, general symmetry measures and more examples, circumradius and inradius in Minkowski spaces. The fourth chapter, finally, is devoted to more recent research, studying the sequence of mean Minkowski measures of symmetry that has been introduced by the author, together with a dual construction. The mean Minkowski measures of symmetry measure, roughly, how far the affine slices of a convex body through interior points are away from simplices on the one hand and from symmetric bodies on the other. Various inequalities and stability estimates are obtained, together with characterizations of simplices. Each chapter has illustrating figures and concludes with exercises, some of them difficult, but partially accompanied by hints and references. The book has a surprising appendix, where certain of the symmetry measures are applied to convex bodies arising as moduli spaces of certain maps from homogeneous Riemannian spaces to spheres. 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. The author points out (based on experience) that the first two chapters have sufficient material for an introductory graduate course in convex geometry.
scientific article; zbMATH DE number 6492282