Convex Bodies

Convex Bodies PDF Author: Rolf Schneider
Publisher: Cambridge University Press
ISBN: 0521352207
Category : Mathematics
Languages : en
Pages : 506

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

Convex Bodies

Convex Bodies PDF Author: Rolf Schneider
Publisher: Cambridge University Press
ISBN: 0521352207
Category : Mathematics
Languages : en
Pages : 506

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

Convex Bodies: The Brunn–Minkowski Theory

Convex Bodies: The Brunn–Minkowski Theory PDF Author: Rolf Schneider
Publisher: Cambridge University Press
ISBN: 1107601010
Category : Mathematics
Languages : en
Pages : 759

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Book Description
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

Convex Bodies: The Brunn–Minkowski Theory PDF Author: Rolf Schneider
Publisher: Cambridge University Press
ISBN: 1107471613
Category : Mathematics
Languages : en
Pages : 752

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Book Description
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.

Lectures on Convex Geometry

Lectures on Convex Geometry PDF Author: Daniel Hug
Publisher: Springer Nature
ISBN: 3030501809
Category : Mathematics
Languages : en
Pages : 300

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Book Description
This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.

Theory of Convex Bodies

Theory of Convex Bodies PDF Author: Tommy Bonnesen
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 192

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Book Description


The Volume of Convex Bodies and Banach Space Geometry

The Volume of Convex Bodies and Banach Space Geometry PDF Author: Gilles Pisier
Publisher: Cambridge University Press
ISBN: 9780521666350
Category : Mathematics
Languages : en
Pages : 270

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

Geometry of Isotropic Convex Bodies

Geometry of Isotropic Convex Bodies PDF Author: Silouanos Brazitikos
Publisher: American Mathematical Soc.
ISBN: 1470414562
Category : Mathematics
Languages : en
Pages : 618

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Book Description
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.

Fourier Analysis in Convex Geometry

Fourier Analysis in Convex Geometry PDF Author: Alexander Koldobsky
Publisher: American Mathematical Soc.
ISBN: 1470419521
Category : Mathematics
Languages : en
Pages : 178

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Book Description
The study of the geometry of convex bodies based on information about sections and projections of these bodies has important applications in many areas of mathematics and science. In this book, a new Fourier analysis approach is discussed. The idea is to express certain geometric properties of bodies in terms of Fourier analysis and to use harmonic analysis methods to solve geometric problems. One of the results discussed in the book is Ball's theorem, establishing the exact upper bound for the -dimensional volume of hyperplane sections of the -dimensional unit cube (it is for each ). Another is the Busemann-Petty problem: if and are two convex origin-symmetric -dimensional bodies and the -dimensional volume of each central hyperplane section of is less than the -dimensional volume of the corresponding section of , is it true that the -dimensional volume of is less than the volume of ? (The answer is positive for and negative for .) The book is suitable for graduate students and researchers interested in geometry, harmonic and functional analysis, and probability. Prerequisites for reading this book include basic real, complex, and functional analysis.

Convexity and Concentration

Convexity and Concentration PDF Author: Eric Carlen
Publisher: Springer
ISBN: 1493970054
Category : Mathematics
Languages : en
Pages : 620

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Book Description
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.

Foundations of Convex Geometry

Foundations of Convex Geometry PDF Author: W. A. Coppel
Publisher: Cambridge University Press
ISBN: 9780521639705
Category : Mathematics
Languages : en
Pages : 236

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Book Description
This book on the foundations of Euclidean geometry aims to present the subject from the point of view of present day mathematics, taking advantage of all the developments since the appearance of Hilbert's classic work. Here real affine space is characterised by a small number of axioms involving points and line segments making the treatment self-contained and thorough, many results being established under weaker hypotheses than usual. The treatment should be totally accessible for final year undergraduates and graduate students, and can also serve as an introduction to other areas of mathematics such as matroids and antimatroids, combinatorial convexity, the theory of polytopes, projective geometry and functional analysis.