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000746188 0247_ $$a10.1007/978-3-319-23733-6$$2doi
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000746188 1001_ $$aToth, Gabor.$$eauthor.
000746188 24510 $$aMeasures of symmetry for convex sets and stability$$h[electronic resource] /$$cby Gabor Toth.
000746188 264_1 $$aCham :$$bSpringer,$$c2015.
000746188 300__ $$a1 online resource (xii, 278 pages) :$$billustrations.
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000746188 4901_ $$aUniversitext,$$x0172-5939
000746188 5050_ $$aFirst Things First on Convex Sets -- Affine Diameters and the Critical Set -- Measures of Stability and Symmetry -- Mean Minkowski Measures.
000746188 506__ $$aAccess limited to authorized users.
000746188 520__ $$aThis 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.
000746188 650_0 $$aMathematics.
000746188 650_0 $$aConvex geometry.
000746188 650_0 $$aDiscrete geometry.
000746188 77608 $$iPrint version:$$z9783319237329
000746188 830_0 $$aUniversitext.
000746188 85280 $$bebk$$hSpringerLink
000746188 85640 $$3SpringerLink$$uhttps://univsouthin.idm.oclc.org/login?url=http://link.springer.com/10.1007/978-3-319-23733-6$$zOnline Access$$91397441.1
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