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Measures of Symmetry for Convex Sets and Stability (Paperback, 1st ed. 2015)
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Measures of Symmetry for Convex Sets and Stability (Paperback, 1st ed. 2015)
Series: Universitext
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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 Grunbaum'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 Caratheodory, 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
Grunbaum'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.
General
Imprint: |
Springer International Publishing AG
|
Country of origin: |
Switzerland |
Series: |
Universitext |
Release date: |
December 2015 |
First published: |
2015 |
Authors: |
Gabor Toth
|
Dimensions: |
235 x 155 x 18mm (L x W x T) |
Format: |
Paperback
|
Pages: |
278 |
Edition: |
1st ed. 2015 |
ISBN-13: |
978-3-319-23732-9 |
Categories: |
Books >
Science & Mathematics >
Mathematics >
Geometry >
Algebraic geometry
|
LSN: |
3-319-23732-2 |
Barcode: |
9783319237329 |
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