By Gabor Toth

ISBN-10: 3319237322

ISBN-13: 9783319237329

This textbook treats vital and comparable concerns in convex geometry: the quantification of symmetry of a convex set―measures of symmetry―and the measure to which convex units that almost reduce such measures of symmetry are themselves approximately symmetric―the phenomenon of balance. through amassing the subject’s center rules and highlights round Grünbaum’s normal concept of degree of symmetry, it paints a coherent photo of the topic, and courses the reader from the fundamentals to the cutting-edge. The exposition takes quite a few paths to leads to order to increase the reader’s snatch of the team spirit of principles, whereas interspersed comments enhance the cloth with a behind-the-scenes view of corollaries and logical connections, substitute proofs, and allied effects from the literature. various illustrations elucidate definitions and key buildings, and over 70 exercises―with tricks and references for the more challenging ones―test and sharpen the reader’s comprehension.

The presentation contains: a simple direction masking 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 conception, a research of convex our bodies of continuing width; proofs of F. John’s ellipsoid theorem; a remedy of the soundness 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, similar to Winternitz degree, the Rogers–Shepard quantity ratio, and Guo’s *L*^{p} -Minkowski degree; a development 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 house of sure uncommon maps from a Riemannian homogeneous house to

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