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Convex Bodies: The Brunn–Minkowski Theory

Convex Bodies: The Brunn–Minkowski Theory
Author: Rolf Schneider
Publisher: Cambridge University Press
Total Pages: 759
Release: 2014
Genre: 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.


Convex Bodies

Convex Bodies
Author: Rolf Schneider
Publisher: Cambridge University Press
Total Pages: 506
Release: 1993-02-25
Genre: Mathematics
ISBN: 0521352207

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


Geometry of Isotropic Convex Bodies

Geometry of Isotropic Convex Bodies
Author: Silouanos Brazitikos
Publisher: American Mathematical Soc.
Total Pages: 618
Release: 2014-04-24
Genre: 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.


The Volume of Convex Bodies and Banach Space Geometry

The Volume of Convex Bodies and Banach Space Geometry
Author: Gilles Pisier
Publisher: Cambridge University Press
Total Pages: 270
Release: 1999-05-27
Genre: Mathematics
ISBN: 9780521666350

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


Convex Bodies and Algebraic Geometry

Convex Bodies and Algebraic Geometry
Author: Tadao Oda
Publisher: Springer
Total Pages: 0
Release: 2012-02-23
Genre: 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.


Convex Sets and Their Applications

Convex Sets and Their Applications
Author: Steven R. Lay
Publisher: Courier Corporation
Total Pages: 260
Release: 2007-01-01
Genre: Mathematics
ISBN: 0486458032

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Suitable for advanced undergraduates and graduate students, this text introduces the broad scope of convexity. It leads students to open questions and unsolved problems, and it highlights diverse applications. Author Steven R. Lay, Professor of Mathematics at Lee University in Tennessee, reinforces his teachings with numerous examples, plus exercises with hints and answers. The first three chapters form the foundation for all that follows, starting with a review of the fundamentals of linear algebra and topology. They also survey the development and applications of relationships between hyperplanes and convex sets. Subsequent chapters are relatively self-contained, each focusing on a particular aspect or application of convex sets. Topics include characterizations of convex sets, polytopes, duality, optimization, and convex functions. Hints, solutions, and references for the exercises appear at the back of the book.


Isoperimetric Inequalities in Unbounded Convex Bodies

Isoperimetric Inequalities in Unbounded Convex Bodies
Author: Gian Paolo Leonardi
Publisher: American Mathematical Society
Total Pages: 86
Release: 2022-04-08
Genre: Mathematics
ISBN: 1470451182

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Convex Bodies: The Brunn–Minkowski Theory

Convex Bodies: The Brunn–Minkowski Theory
Author: Rolf Schneider
Publisher: Cambridge University Press
Total Pages: 752
Release: 2013-10-31
Genre: 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.