© 2015

Measures of Symmetry for Convex Sets and Stability


Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Gabor Toth
    Pages 1-46
  3. Gabor Toth
    Pages 47-94
  4. Gabor Toth
    Pages 95-158
  5. Gabor Toth
    Pages 159-241
  6. Back Matter
    Pages 243-278

About this book


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.


Banach-Mazur metric Groemer stability of the Brunn-Minkowsi inequality Gruenbaum's general notion of measure of symmetry Helly's theorem John's ellipsoid theorem Minkowski measures basic metric invariants convex geometry critical sets mean Minkowski measures stability symmetry

Authors and affiliations

  1. 1.Department of Mathematical SciencesRutgers UniversityCamdenUSA

About the authors

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.

Bibliographic information

  • Book Title Measures of Symmetry for Convex Sets and Stability
  • Authors Gabor Toth
  • Series Title Universitext
  • Series Abbreviated Title Universitext
  • DOI
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-23732-9
  • eBook ISBN 978-3-319-23733-6
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages XII, 278
  • Number of Illustrations 64 b/w illustrations, 1 illustrations in colour
  • Topics Convex and Discrete Geometry
  • Buy this book on publisher's site


“The book under review is a graduate-level textbook on convexity, which presents the topic from a new and interesting point of view. … The book offers the reader a new approach to the study of convexity, focusing on the important topics of measures of symmetry and stability. It moves from the very beginning background to recent research, and therefore both students and researchers can benefit from it.” (María A. Hernández Cifre, Mathematical Reviews, December, 2016) 

“This is a graduate-level textbook on convex geometry in finite-dimensional Euclidean spaces, which has some interesting special features. … Each chapter has illustrating figures and concludes with exercises … . The book has a surprising appendix, where certain of the symmetry measures are applied to convex bodies … . 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.” (Rolf Schneider, zbMATH 1335.52002, 2016)