[PDF] Convex Bodies And Algebraic Geometry eBook

Author : Tadao Oda
Publisher : Springer
Page : 234 pages
File Size : 38,48 MB
Release : 1988
Category : Mathematics
ISBN :

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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 : Günter Ewald
Publisher : Springer Science & Business Media
Page : 378 pages
File Size : 17,11 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 1461240441

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The book is an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry, and to the connections between these fields, known as the theory of toric varieties. The first part of the book covers the theory of polytopes and provides large parts of the mathematical background of linear optimization and of the geometrical aspects in computer science. The second part introduces toric varieties in an elementary way.

Author : Bozzano G Luisa
Publisher : Elsevier
Page : 803 pages
File Size : 36,23 MB
Release : 2014-06-28
Category : Mathematics
ISBN : 0080934390

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Handbook of Convex Geometry, Volume A offers a survey of convex geometry and its many ramifications and relations with other areas of mathematics, including convexity, geometric inequalities, and convex sets. The selection first offers information on the history of convexity, characterizations of convex sets, and mixed volumes. Topics include elementary convexity, equality in the Aleksandrov-Fenchel inequality, mixed surface area measures, characteristic properties of convex sets in analysis and differential geometry, and extensions of the notion of a convex set. The text then reviews the standard isoperimetric theorem and stability of geometric inequalities. The manuscript takes a look at selected affine isoperimetric inequalities, extremum problems for convex discs and polyhedra, and rigidity. Discussions focus on include infinitesimal and static rigidity related to surfaces, isoperimetric problem for convex polyhedral, bounds for the volume of a convex polyhedron, curvature image inequality, Busemann intersection inequality and its relatives, and Petty projection inequality. The book then tackles geometric algorithms, convexity and discrete optimization, mathematical programming and convex geometry, and the combinatorial aspects of convex polytopes. The selection is a valuable source of data for mathematicians and researchers interested in convex geometry.

Author : Maria Moszynska
Publisher : Springer Science & Business Media
Page : 223 pages
File Size : 50,35 MB
Release : 2006-11-24
Category : Mathematics
ISBN : 0817644512

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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 : Rolf Schneider
Publisher : Cambridge University Press
Page : 759 pages
File Size : 46,38 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 : 47,13 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 : Grigoriy Blekherman
Publisher : SIAM
Page : 487 pages
File Size : 38,86 MB
Release : 2013-03-21
Category : Mathematics
ISBN : 1611972280

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An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.

Author : Peter M. Gruber
Publisher : North Holland
Page : 774 pages
File Size : 41,59 MB
Release : 1993-08-24
Category : Mathematics
ISBN :

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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 : John Wiley & Sons
Page : 280 pages
File Size : 14,32 MB
Release : 1979-05
Category : Mathematics
ISBN :

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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.

Author : Keith M. Ball
Publisher : Cambridge University Press
Page : 260 pages
File Size : 35,23 MB
Release : 1999-01-28
Category : Mathematics
ISBN : 9780521642590

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Articles on classical convex geometry, geometric functional analysis, computational geometry, and related areas of harmonic analysis, first published in 1999.