By Gabor Toth
This textbook treats very important and similar concerns 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. by means of accumulating the subject’s center rules and highlights round Grünbaum’s basic thought of degree of symmetry, it paints a coherent photo of the topic, and publications the reader from the fundamentals to the cutting-edge. The exposition takes a number of paths to leads to order to boost the reader’s seize of the solidarity of principles, whereas interspersed comments improve the fabric with a behind-the-scenes view of corollaries and logical connections, substitute proofs, and allied effects from the literature. a variety 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 contains: a uncomplicated path overlaying foundational notions in convex geometry, the 3 pillars of the combinatorial concept (the theorems of Carathéodory, Radon, and Helly), severe units and Minkowski degree, the Minkowski–Radon inequality, and, to demonstrate the overall conception, 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, akin to Winternitz degree, the Rogers–Shepard quantity ratio, and Guo’s Lp -Minkowski degree; a building by means of the writer of a brand new series of measures of symmetry, the kth suggest Minkowski degree; and finally, an interesting software to the moduli area of sure unique maps from a Riemannian homogeneous area to
spheres—illustrating the huge mathematical relevance of the book’s subject.
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