[PDF] Theory Of Convex Bodies eBook

Author : Rolf Schneider
Publisher : Cambridge University Press
Page : 759 pages
File Size : 48,81 MB
Release : 2014
Category : Mathematics
ISBN : 1107601010

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A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research.

Author : Silouanos Brazitikos
Publisher : American Mathematical Soc.
Page : 618 pages
File Size : 17,85 MB
Release : 2014-04-24
Category : Mathematics
ISBN : 1470414562

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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 : Tadao Oda
Publisher : Springer
Page : 0 pages
File Size : 30,38 MB
Release : 2012-02-23
Category : Mathematics
ISBN : 9783642725494

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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 : Gilles Pisier
Publisher : Cambridge University Press
Page : 270 pages
File Size : 28,48 MB
Release : 1999-05-27
Category : Mathematics
ISBN : 9780521666350

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A self-contained presentation of results relating the volume of convex bodies and Banach space geometry.

Author : Rolf Schneider
Publisher : Cambridge University Press
Page : 752 pages
File Size : 26,78 MB
Release : 2013-10-31
Category : Mathematics
ISBN : 1107471613

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At the heart of this monograph is the Brunn–Minkowski theory, which can be used to great effect in studying such ideas as volume and surface area and their generalizations. In particular, the notions of mixed volume and mixed area measure arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered here in detail. The author presents a comprehensive introduction to convex bodies, including full proofs for some deeper theorems. The book provides hints and pointers to connections with other fields and an exhaustive reference list. This second edition has been considerably expanded to reflect the rapid developments of the past two decades. It includes new chapters on valuations on convex bodies, on extensions like the Lp Brunn–Minkowski theory, and on affine constructions and inequalities. There are also many supplements and updates to the original chapters, and a substantial expansion of chapter notes and references.

Author : Bozzano G Luisa
Publisher : Elsevier
Page : 769 pages
File Size : 37,54 MB
Release : 2014-06-28
Category : Mathematics
ISBN : 0080934404

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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 : Paul J. Kelly
Publisher :
Page : 0 pages
File Size : 41,24 MB
Release : 2009
Category : Convex bodies
ISBN : 9780486469805

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This text assumes no prerequisites, offering an easy-to-read treatment with simple notation and clear, complete proofs. From motivation to definition, its explanations feature concrete examples and theorems. 1979 edition.