Umfang:
Online-Ressource (XII, 278 p. 65 illus., 1 illus. in color, online resource)
Ausgabe:
1st ed. 2015
ISBN:
9783319237336
Serie:
Universitext
Inhalt:
First Things First on Convex Sets -- Affine Diameters and the Critical Set -- Measures of Stability and Symmetry -- Mean Minkowski Measures
Inhalt:
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 the book’s subject
Anmerkung:
First Things First on Convex SetsAffine Diameters and the Critical Set -- Measures of Stability and Symmetry -- Mean Minkowski Measures.
Weitere Ausg.:
ISBN 9783319237329
Weitere Ausg.:
Druckausg. Toth, Gabor, 1954 - Measures of symmetry for convex sets and stability Cham : Springer, 2015 ISBN 9783319237329
Weitere Ausg.:
Erscheint auch als Druck-Ausgabe Toth, Gabor, 1954 - Measures of symmetry for convex sets and stability Cham [u.a.] : Springer, 2016 ISBN 9783319237329
Weitere Ausg.:
ISBN 3319237322
Sprache:
Englisch
Fachgebiete:
Mathematik
Schlagwort(e):
Konvexe Menge
DOI:
10.1007/978-3-319-23733-6
URL:
Volltext
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