Convex Bodies: The Brunn–Minkowski Theory

Author: Rolf Schneider

Publisher: Cambridge University Press

ISBN: 1107601010

Category: Mathematics

Page: 760

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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: The Brunn–Minkowski Theory

Author: Rolf Schneider

Publisher: Cambridge University Press

ISBN: 1107471613

Category: Mathematics

Page: 752

View: 3690

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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.

Geometric Tomography

Author: Richard J. Gardner

Publisher: Cambridge University Press

ISBN: 0521866804

Category: Mathematics

Page: 492

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A comprehensive, rigorous treatment, with 66 unsolved problems, over 70 illustrations, and over 800 references.

Convex Bodies

The Brunn-Minkowski Theory

Author: Rolf Schneider

Publisher: Cambridge University Press

ISBN: 9780521352208

Category: Mathematics

Page: 490

View: 2781

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

A Course in Convexity

Author: Alexander Barvinok

Publisher: American Mathematical Soc.

ISBN: 0821829688

Category: Mathematics

Page: 366

View: 9473

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Convexity is a simple idea that manifests itself in a surprising variety of places. This fertile field has an immensely rich structure and numerous applications. Barvinok demonstrates that simplicity, intuitive appeal, and the universality of applications make teaching (and learning) convexity a gratifying experience. The book will benefit both teacher and student: It is easy to understand, entertaining to the reader, and includes many exercises that vary in degree of difficulty. Overall, the author demonstrates the power of a few simple unifying principles in a variety of pure and applied problems. The prerequisites are minimal amounts of linear algebra, analysis, and elementary topology, plus basic computational skills. Portions of the book could be used by advanced undergraduates. As a whole, it is designed for graduate students interested in mathematical methods, computer science, electrical engineering, and operations research. The book will also be of interest to research mathematicians, who will find some results that are recent, some that are new, and many known results that are discussed from a new perspective.

Polytopes

Abstract, Convex and Computational

Author: Tibor Bisztriczky,Peter McMullen,Rolf Schneider,Asia Ivic Weiss

Publisher: Springer Science & Business Media

ISBN: 9401109249

Category: Mathematics

Page: 507

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The aim of this volume is to reinforce the interaction between the three main branches (abstract, convex and computational) of the theory of polytopes. The articles include contributions from many of the leading experts in the field, and their topics of concern are expositions of recent results and in-depth analyses of the development (past and future) of the subject. The subject matter of the book ranges from algorithms for assignment and transportation problems to the introduction of a geometric theory of polyhedra which need not be convex. With polytopes as the main topic of interest, there are articles on realizations, classifications, Eulerian posets, polyhedral subdivisions, generalized stress, the Brunn--Minkowski theory, asymptotic approximations and the computation of volumes and mixed volumes. For researchers in applied and computational convexity, convex geometry and discrete geometry at the graduate and postgraduate levels.

Geometry of Isotropic Convex Bodies

Author: Silouanos Brazitikos,Apostolos Giannopoulos,Petros Valettas,Beatrice-Helen Vritsiou

Publisher: American Mathematical Soc.

ISBN: 1470414562

Category: Mathematics

Page: 594

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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.

Minkowski Geometry

Author: A. C. Thompson

Publisher: Cambridge University Press

ISBN: 9780521404723

Category: Mathematics

Page: 346

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This is a comprehensive treatment of Minkowski geometry. The author begins by describing the fundamental metric properties and the topological properties of existence of Minkowski space. This is followed by a treatment of two-dimensional spaces and characterizations of Euclidean space among normed spaces. The central three chapters present the theory of area and volume in normed spaces--a fascinating geometrical interplay among the various roles of the ball in Euclidean space. Later chapters deal with trigonometry and differential geometry in Minkowski spaces. The book ends with a brief look at J. J. Schaffer's ideas on the intrinsic geometry of the unit sphere.

Convexity and Concentration

Author: Eric Carlen,Mokshay Madiman,Elisabeth M. Werner

Publisher: Springer

ISBN: 1493970054

Category: Mathematics

Page: 626

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This volume presents some of the research topics discussed at the 2014-2015 Annual Thematic Program Discrete Structures: Analysis and Applications at the Institute of Mathematics and its Applications during the Spring 2015 where geometric analysis, convex geometry and concentration phenomena were the focus. Leading experts have written surveys of research problems, making state of the art results more conveniently and widely available. The volume is organized into two parts. Part I contains those contributions that focus primarily on problems motivated by probability theory, while Part II contains those contributions that focus primarily on problems motivated by convex geometry and geometric analysis. This book will be of use to those who research convex geometry, geometric analysis and probability directly or apply such methods in other fields.

Tensor Valuations and Their Applications in Stochastic Geometry and Imaging

Author: Eva B. Vedel Jensen,Markus Kiderlen

Publisher: Springer

ISBN: 3319519514

Category: Mathematics

Page: 462

View: 5953

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The purpose of this volume is to give an up-to-date introduction to tensor valuations and their applications. Starting with classical results concerning scalar-valued valuations on the families of convex bodies and convex polytopes, it proceeds to the modern theory of tensor valuations. Product and Fourier-type transforms are introduced and various integral formulae are derived. New and well-known results are presented, together with generalizations in several directions, including extensions to the non-Euclidean setting and to non-convex sets. A variety of applications of tensor valuations to models in stochastic geometry, to local stereology and to imaging are also discussed.

Algebraic and Geometric Combinatorics

Euroconference in Mathematics : Algebraic and Geometric Combinatorics, August 20-26, 2005, Anogia, Crete, Greece

Author: Christos A. Athanasiadis

Publisher: American Mathematical Soc.

ISBN: 0821840800

Category: Mathematics

Page: 324

View: 443

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This volume contains original research and survey articles stemming from the Euroconference ""Algebraic and Geometric Combinatorics"". The papers discuss a wide range of problems that illustrate interactions of combinatorics with other branches of mathematics, such as commutative algebra, algebraic geometry, convex and discrete geometry, enumerative geometry, and topology of complexes and partially ordered sets. Among the topics covered are combinatorics of polytopes, lattice polytopes, triangulations and subdivisions, Cohen-Macaulay cell complexes, monomial ideals, geometry of toric surfaces, groupoids in combinatorics, Kazhdan-Lusztig combinatorics, and graph colorings. This book is aimed at researchers and graduate students interested in various aspects of modern combinatorial theories.

Convex and Discrete Geometry

Author: Peter Gruber

Publisher: Springer Science & Business Media

ISBN: 3540711333

Category: Mathematics

Page: 580

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Convex and Discrete Geometry is an area of mathematics situated between analysis, geometry and discrete mathematics with numerous relations to other subdisciplines. This book provides a comprehensive overview of major results, methods and ideas of convex and discrete geometry and its applications. Besides being a graduate-level introduction to the field, it is a practical source of information and orientation for convex geometers, and useful to people working in the applied fields.

Concentration, Functional Inequalities, and Isoperimetry

International Workshop on Concentration, Functional Inequalities, and Isoperimetry, October 29-November 1, 2009, Florida Atlantic University, Boca Raton, Florida

Author: Christian Houdré

Publisher: American Mathematical Soc.

ISBN: 0821849719

Category: Mathematics

Page: 211

View: 896

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The volume contains the proceedings of the international workshop on Concentration, Functional Inequalities and Isoperimetry, held at Florida Atlantic University in Boca Raton, Florida, from October 29-November 1, 2009. The interactions between concentration, isoperimetry and functional inequalities have led to many significant advances in functional analysis and probability theory. Important progress has also taken place in combinatorics, geometry, harmonic analysis and mathematical physics, to name but a few fields, with recent new applications in random matrices and information theory. This book should appeal to graduate students and researchers interested in the fascinating interplay between analysis, probability, and geometry.

Encyclopedia of Mathematics

Author: James Stuart Tanton

Publisher: Infobase Publishing

ISBN: 1438110081

Category: Mathematics

Page: 577

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Encyclopedia of Mathematics is a comprehensive one-volume encyclopedia designed for high school through early college students. More than 1,000 entries, numerous essays, and more than 125 photographs and illustrations cover the principal areas and issues that characterize this "new" area of science. This valuable resource unites disparate ideas and provides the meaning, history, context, and relevance behind each one. The easy-to-use format makes finding straightforward and natural answers to questions within arithmetic simple. Encyclopedia of Mathematics also gives historical context to mathematical concepts, with entries discussing ancient Arabic, Babylonian, Chinese, Egyptian, Greek, Hindu, and Mayan mathematics, as well as entries providing biographical descriptions of important people in the development of mathematics.

Asymptotic Geometric Analysis, Part I

Author: Shiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman

Publisher: American Mathematical Soc.

ISBN: 1470421933

Category: Functional analysis

Page: 451

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The authors present the theory of asymptotic geometric analysis, a field which lies on the border between geometry and functional analysis. In this field, isometric problems that are typical for geometry in low dimensions are substituted by an "isomorphic" point of view, and an asymptotic approach (as dimension tends to infinity) is introduced. Geometry and analysis meet here in a non-trivial way. Basic examples of geometric inequalities in isomorphic form which are encountered in the book are the "isomorphic isoperimetric inequalities" which led to the discovery of the "concentration phenomenon", one of the most powerful tools of the theory, responsible for many counterintuitive results. A central theme in this book is the interaction of randomness and pattern. At first glance, life in high dimension seems to mean the existence of multiple "possibilities", so one may expect an increase in the diversity and complexity as dimension increases. However, the concentration of measure and effects caused by convexity show that this diversity is compensated and order and patterns are created for arbitrary convex bodies in the mixture caused by high dimensionality. The book is intended for graduate students and researchers who want to learn about this exciting subject. Among the topics covered in the book are convexity, concentration phenomena, covering numbers, Dvoretzky-type theorems, volume distribution in convex bodies, and more.

Convex Polyhedra

Author: A.D. Alexandrov

Publisher: Springer Science & Business Media

ISBN: 3540263403

Category: Mathematics

Page: 542

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This classic geometry text explores the theory of 3-dimensional convex polyhedra in a unique fashion, with exceptional detail. Vital and clearly written, the book includes the basics of convex polyhedra and collects the most general existence theorems for convex polyhedra that are proved by a new and unified method. This edition includes a comprehensive bibliography by V.A. Zalgaller, and related papers as supplements to the original text.

Introduction to Geometric Probability

Author: Daniel A. Klain,Gian-Carlo Rota

Publisher: Cambridge University Press

ISBN: 9780521596541

Category: Mathematics

Page: 178

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The basic ideas of the subject and the analogues with enumerative combinatorics are described and exploited.

Equivalents of the Riemann Hypothesis: Volume 2, Analytic Equivalents

Author: Kevin Broughan

Publisher: Cambridge University Press

ISBN: 1108187021

Category: Mathematics

Page: N.A

View: 4130

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The Riemann hypothesis (RH) is perhaps the most important outstanding problem in mathematics. This two-volume text presents the main known equivalents to RH using analytic and computational methods. The book is gentle on the reader with definitions repeated, proofs split into logical sections, and graphical descriptions of the relations between different results. It also includes extensive tables, supplementary computational tools, and open problems suitable for research. Accompanying software is free to download. These books will interest mathematicians who wish to update their knowledge, graduate and senior undergraduate students seeking accessible research problems in number theory, and others who want to explore and extend results computationally. Each volume can be read independently. Volume 1 presents classical and modern arithmetic equivalents to RH, with some analytic methods. Volume 2 covers equivalences with a strong analytic orientation, supported by an extensive set of appendices containing fully developed proofs.

Handbook of Categorical Algebra: Volume 1, Basic Category Theory

Author: Francis Borceux

Publisher: Cambridge University Press

ISBN: 9780521441780

Category: Mathematics

Page: 345

View: 9365

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The Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence, with the first being essentially self-contained, and are accessible to graduate students with a good background in mathematics. In particular, Volume 1, which is devoted to general concepts, can be used for advanced undergraduate courses on category theory.

Theory of convex bodies

Author: Tommy Bonnesen,Werner Fenchel

Publisher: B C S Associates

ISBN: N.A

Category: Mathematics

Page: 172

View: 1335

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Growing up as the oldest son of King Arthur, sixteen-year-old Zeben experiences his first tournament, a clash with traitorous knights, and a secret alliance with Merlin to save the royal family from crippling dissension.