Geometry Of Isotropic Convex Bodies
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Author | : Silouanos Brazitikos |
Publisher | : American Mathematical Soc. |
Total Pages | : 618 |
Release | : 2014-04-24 |
Genre | : Mathematics |
ISBN | : 1470414562 |
The study of high-dimensional convex bodies from a geometric and analytic point of view, with an emphasis on the dependence of various parameters on the dimension stands at the intersection of classical convex geometry and the local theory of Banach spaces. It is also closely linked to many other fields, such as probability theory, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics. It is now understood that the convexity assumption forces most of the volume of a high-dimensional convex body to be concentrated in some canonical way and the main question is whether, under some natural normalization, the answer to many fundamental questions should be independent of the dimension. The aim of this book is to introduce a number of well-known questions regarding the distribution of volume in high-dimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the Kannan-Lovász-Simonovits conjecture. This book provides a self-contained and up to date account of the progress that has been made in the last fifteen years.
Author | : Bozzano G Luisa |
Publisher | : Elsevier |
Total Pages | : 769 |
Release | : 2014-06-28 |
Genre | : Mathematics |
ISBN | : 0080934404 |
Handbook of Convex Geometry, Volume B offers a survey of convex geometry and its many ramifications and connections with other fields of mathematics, including convexity, lattices, crystallography, and convex functions. The selection first offers information on the geometry of numbers, lattice points, and packing and covering with convex sets. Discussions focus on packing in non-Euclidean spaces, problems in the Euclidean plane, general convex bodies, computational complexity of lattice point problem, centrally symmetric convex bodies, reduction theory, and lattices and the space of lattices. The text then examines finite packing and covering and tilings, including plane tilings, monohedral tilings, bin packing, and sausage problems. The manuscript takes a look at valuations and dissections, geometric crystallography, convexity and differential geometry, and convex functions. Topics include differentiability, inequalities, uniqueness theorems for convex hypersurfaces, mixed discriminants and mixed volumes, differential geometric characterization of convexity, reduction of quadratic forms, and finite groups of symmetry operations. The selection is a dependable source of data for mathematicians and researchers interested in convex geometry.
Author | : Maria Moszynska |
Publisher | : Springer Science & Business Media |
Total Pages | : 223 |
Release | : 2006-11-24 |
Genre | : Mathematics |
ISBN | : 0817644512 |
Examines in detail those topics in convex geometry that are concerned with Euclidean space Enriched by numerous examples, illustrations, and exercises, with a good bibliography and index Requires only a basic knowledge of geometry, linear algebra, analysis, topology, and measure theory Can be used for graduates courses or seminars in convex geometry, geometric and convex combinatorics, and convex analysis and optimization
Author | : Keith M. Ball |
Publisher | : Cambridge University Press |
Total Pages | : 260 |
Release | : 1999-01-28 |
Genre | : Mathematics |
ISBN | : 9780521642590 |
Articles on classical convex geometry, geometric functional analysis, computational geometry, and related areas of harmonic analysis, first published in 1999.
Author | : Daniel Hug |
Publisher | : Springer Nature |
Total Pages | : 287 |
Release | : 2020-08-27 |
Genre | : Mathematics |
ISBN | : 3030501809 |
This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.
Author | : Alexander Koldobsky |
Publisher | : American Mathematical Soc. |
Total Pages | : 128 |
Release | : |
Genre | : Mathematics |
ISBN | : 9780821883358 |
"The book is written in the form of lectures accessible to graduate students. This approach allows the reader to clearly see the main ideas behind the method, rather than to dwell on technical difficulties. The book also contains discussions of the most recent advances in the subject. The first section of each lecture is a snapshot of that lecture. By reading each of these sections first, novices can gain an overview of the subject, then return to the full text for more details."--BOOK JACKET.
Author | : Rolf Schneider |
Publisher | : Cambridge University Press |
Total Pages | : 759 |
Release | : 2014 |
Genre | : Mathematics |
ISBN | : 1107601010 |
A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research.
Author | : Tadao Oda |
Publisher | : Springer |
Total Pages | : 234 |
Release | : 1988 |
Genre | : Mathematics |
ISBN | : |
The theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces. This book is a unified up-to-date survey of the various results and interesting applications found since toric varieties were introduced in the early 1970's. It is an updated and corrected English edition of the author's book in Japanese published by Kinokuniya, Tokyo in 1985. Toric varieties are here treated as complex analytic spaces. Without assuming much prior knowledge of algebraic geometry, the author shows how elementary convex figures give rise to interesting complex analytic spaces. Easily visualized convex geometry is then used to describe algebraic geometry for these spaces, such as line bundles, projectivity, automorphism groups, birational transformations, differential forms and Mori's theory. Hence this book might serve as an accessible introduction to current algebraic geometry. Conversely, the algebraic geometry of toric varieties gives new insight into continued fractions as well as their higher-dimensional analogues, the isoperimetric problem and other questions on convex bodies. Relevant results on convex geometry are collected together in the appendix.
Author | : Kurt Leichtweiß |
Publisher | : Wiley-VCH |
Total Pages | : 0 |
Release | : 1999-01-12 |
Genre | : Science |
ISBN | : 9783527402618 |
The theory of convex bodies is nowadays an important independent topic of geometry. The author discusses the equiaffine geometry and differential geometry of convex bodies and convex surfaces and especially stresses analogies to classical Euclidean differential geometry. These theories are illustrated by practical applications in areas such as shipbuilding. He offers an accessible introduction to the latest developments in the subject.
Author | : Paul J. Kelly |
Publisher | : John Wiley & Sons |
Total Pages | : 280 |
Release | : 1979-05 |
Genre | : Mathematics |
ISBN | : |
Convex body theory offers important applications in probability and statistics, combinatorial mathematics, and optimization theory. Although this text's setting and central issues are geometric in nature, it stresses the interplay of concepts and methods from topology, analysis, and linear and affine algebra. From motivation to definition, the authors present concrete examples and theorems that identify convex bodies and surfaces and establish their basic properties. The easy-to-read treatment employs simple notation and clear, complete proofs. Introductory chapters establish the basics of metric topology and the structure of Euclidean n-space. Subsequent chapters apply this background to the dimension, basic structure, and general geometry of convex bodies and surfaces. Concluding chapters illustrate nonintuitive results to offer students a perspective on the wide range of problems and applications in convex body theory.