Convex Bodies: The Brunn–Minkowski Theory

Convex Bodies: The Brunn–Minkowski Theory
Author :
Publisher : Cambridge University Press
Total Pages : 759
Release :
ISBN-10 : 9781107601017
ISBN-13 : 1107601010
Rating : 4/5 (17 Downloads)

Synopsis Convex Bodies: The Brunn–Minkowski Theory by : Rolf Schneider

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

Geometry of Isotropic Convex Bodies

Geometry of Isotropic Convex Bodies
Author :
Publisher : American Mathematical Soc.
Total Pages : 618
Release :
ISBN-10 : 9781470414566
ISBN-13 : 1470414562
Rating : 4/5 (66 Downloads)

Synopsis Geometry of Isotropic Convex Bodies by : Silouanos Brazitikos

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.

Convex Bodies and Algebraic Geometry

Convex Bodies and Algebraic Geometry
Author :
Publisher : Springer
Total Pages : 0
Release :
ISBN-10 : 364272549X
ISBN-13 : 9783642725494
Rating : 4/5 (9X Downloads)

Synopsis Convex Bodies and Algebraic Geometry by : Tadao Oda

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.

The Volume of Convex Bodies and Banach Space Geometry

The Volume of Convex Bodies and Banach Space Geometry
Author :
Publisher : Cambridge University Press
Total Pages : 270
Release :
ISBN-10 : 052166635X
ISBN-13 : 9780521666350
Rating : 4/5 (5X Downloads)

Synopsis The Volume of Convex Bodies and Banach Space Geometry by : Gilles Pisier

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

Convex Bodies

Convex Bodies
Author :
Publisher : Cambridge University Press
Total Pages : 506
Release :
ISBN-10 : 9780521352208
ISBN-13 : 0521352207
Rating : 4/5 (08 Downloads)

Synopsis Convex Bodies by : Rolf Schneider

A comprehensive introduction to convex bodies giving full proofs for some deeper theorems which have never previously been brought together.

Isoperimetric Inequalities in Unbounded Convex Bodies

Isoperimetric Inequalities in Unbounded Convex Bodies
Author :
Publisher : American Mathematical Society
Total Pages : 86
Release :
ISBN-10 : 9781470451189
ISBN-13 : 1470451182
Rating : 4/5 (89 Downloads)

Synopsis Isoperimetric Inequalities in Unbounded Convex Bodies by : Gian Paolo Leonardi

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Convex Bodies: The Brunn–Minkowski Theory

Convex Bodies: The Brunn–Minkowski Theory
Author :
Publisher : Cambridge University Press
Total Pages : 752
Release :
ISBN-10 : 9781107471610
ISBN-13 : 1107471613
Rating : 4/5 (10 Downloads)

Synopsis Convex Bodies: The Brunn–Minkowski Theory by : Rolf Schneider

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.

Geometry and Convexity

Geometry and Convexity
Author :
Publisher :
Total Pages : 0
Release :
ISBN-10 : 0486469808
ISBN-13 : 9780486469805
Rating : 4/5 (08 Downloads)

Synopsis Geometry and Convexity by : Paul J. Kelly

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.