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 thebook's subject.
