By Gabor Toth

This textbook treats very important and similar issues in convex geometry: the quantification of symmetry of a convex set―measures of symmetry―and the measure to which convex units that just about reduce such measures of symmetry are themselves approximately symmetric―the phenomenon of balance. through collecting the subject’s middle principles and highlights round Grünbaum’s normal thought of degree of symmetry, it paints a coherent photograph of the topic, and publications the reader from the fundamentals to the cutting-edge. The exposition takes quite a few paths to ends up in order to enhance the reader’s take hold of of the team spirit of principles, whereas interspersed feedback enhance the cloth with a behind-the-scenes view of corollaries and logical connections, replacement proofs, and allied effects from the literature. a number of illustrations elucidate definitions and key buildings, and over 70 exercises―with tricks and references for the tougher ones―test and sharpen the reader’s comprehension.

The presentation comprises: a easy direction protecting foundational notions in convex geometry, the 3 pillars of the combinatorial idea (the theorems of Carathéodory, Radon, and Helly), severe units and Minkowski degree, the Minkowski–Radon inequality, and, to demonstrate the final concept, a research of convex our bodies of continuous width; proofs of F. John’s ellipsoid theorem; a therapy of the steadiness of Minkowski degree, the Banach–Mazur metric, and Groemer’s balance estimate for the Brunn–Minkowski inequality; very important specializations of Grünbaum’s summary degree of symmetry, reminiscent of Winternitz degree, the Rogers–Shepard quantity ratio, and Guo’s *L ^{p}* -Minkowski degree; a building by way of the writer of a brand new series of measures of symmetry, the

*k*th suggest Minkowski degree; and finally, an fascinating software to the moduli area of convinced amazing maps from a Riemannian homogeneous house to

spheres―illustrating the large mathematical relevance of the book’s subject.