Foundations Of Convex Geometry

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Foundations of Convex Geometry

Author : W. A. Coppel
Publisher : Cambridge University Press
ISBN 13 : 9780521639705
Total Pages : 236 pages
Book Rating : 4.6/5 (397 download)

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Book Synopsis Foundations of Convex Geometry by : W. A. Coppel

Download or read book Foundations of Convex Geometry written by W. A. Coppel and published by Cambridge University Press. This book was released on 1998-03-05 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Convex and Discrete Geometry

Author : Peter M. Gruber
Publisher : Springer Science & Business Media
ISBN 13 : 3540711333
Total Pages : 590 pages
Book Rating : 4.5/5 (47 download)

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Book Synopsis Convex and Discrete Geometry by : Peter M. Gruber

Download or read book Convex and Discrete Geometry written by Peter M. Gruber and published by Springer Science & Business Media. This book was released on 2007-05-17 with total page 590 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Geometry of Convex Sets

Author : I. E. Leonard
Publisher : John Wiley & Sons
ISBN 13 : 1119022665
Total Pages : 340 pages
Book Rating : 4.1/5 (19 download)

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Book Synopsis Geometry of Convex Sets by : I. E. Leonard

Download or read book Geometry of Convex Sets written by I. E. Leonard and published by John Wiley & Sons. This book was released on 2015-11-02 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: A gentle introduction to the geometry of convex sets in n-dimensional space Geometry of Convex Sets begins with basic definitions of the concepts of vector addition and scalar multiplication and then defines the notion of convexity for subsets of n-dimensional space. Many properties of convex sets can be discovered using just the linear structure. However, for more interesting results, it is necessary to introduce the notion of distance in order to discuss open sets, closed sets, bounded sets, and compact sets. The book illustrates the interplay between these linear and topological concepts, which makes the notion of convexity so interesting. Thoroughly class-tested, the book discusses topology and convexity in the context of normed linear spaces, specifically with a norm topology on an n-dimensional space. Geometry of Convex Sets also features: An introduction to n-dimensional geometry including points; lines; vectors; distance; norms; inner products; orthogonality; convexity; hyperplanes; and linear functionals Coverage of n-dimensional norm topology including interior points and open sets; accumulation points and closed sets; boundary points and closed sets; compact subsets of n-dimensional space; completeness of n-dimensional space; sequences; equivalent norms; distance between sets; and support hyperplanes · Basic properties of convex sets; convex hulls; interior and closure of convex sets; closed convex hulls; accessibility lemma; regularity of convex sets; affine hulls; flats or affine subspaces; affine basis theorem; separation theorems; extreme points of convex sets; supporting hyperplanes and extreme points; existence of extreme points; Krein–Milman theorem; polyhedral sets and polytopes; and Birkhoff’s theorem on doubly stochastic matrices Discussions of Helly’s theorem; the Art Gallery theorem; Vincensini’s problem; Hadwiger’s theorems; theorems of Radon and Caratheodory; Kirchberger’s theorem; Helly-type theorems for circles; covering problems; piercing problems; sets of constant width; Reuleaux triangles; Barbier’s theorem; and Borsuk’s problem Geometry of Convex Sets is a useful textbook for upper-undergraduate level courses in geometry of convex sets and is essential for graduate-level courses in convex analysis. An excellent reference for academics and readers interested in learning the various applications of convex geometry, the book is also appropriate for teachers who would like to convey a better understanding and appreciation of the field to students. I. E. Leonard, PhD, was a contract lecturer in the Department of Mathematical and Statistical Sciences at the University of Alberta. The author of over 15 peer-reviewed journal articles, he is a technical editor for the Canadian Applied Mathematical Quarterly journal. J. E. Lewis, PhD, is Professor Emeritus in the Department of Mathematical Sciences at the University of Alberta. He was the recipient of the Faculty of Science Award for Excellence in Teaching in 2004 as well as the PIMS Education Prize in 2002.

Geometry and Convexity

Author : Paul J. Kelly
Publisher :
ISBN 13 : 9780486469805
Total Pages : 0 pages
Book Rating : 4.4/5 (698 download)

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Book Synopsis Geometry and Convexity by : Paul J. Kelly

Download or read book Geometry and Convexity written by Paul J. Kelly and published by . This book was released on 2009 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Convex Sets and Their Applications

Author : Steven R. Lay
Publisher : Courier Corporation
ISBN 13 : 0486458032
Total Pages : 260 pages
Book Rating : 4.4/5 (864 download)

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Book Synopsis Convex Sets and Their Applications by : Steven R. Lay

Download or read book Convex Sets and Their Applications written by Steven R. Lay and published by Courier Corporation. This book was released on 2007-01-01 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Convex Cones

Author : Rolf Schneider
Publisher : Springer Nature
ISBN 13 : 3031151275
Total Pages : 352 pages
Book Rating : 4.0/5 (311 download)

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Book Synopsis Convex Cones by : Rolf Schneider

Download or read book Convex Cones written by Rolf Schneider and published by Springer Nature. This book was released on 2022-09-21 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides the foundations for geometric applications of convex cones and presents selected examples from a wide range of topics, including polytope theory, stochastic geometry, and Brunn–Minkowski theory. Giving an introduction to convex cones, it describes their most important geometric functionals, such as conic intrinsic volumes and Grassmann angles, and develops general versions of the relevant formulas, namely the Steiner formula and kinematic formula. In recent years questions related to convex cones have arisen in applied mathematics, involving, for example, properties of random cones and their non-trivial intersections. The prerequisites for this work, such as integral geometric formulas and results on conic intrinsic volumes, were previously scattered throughout the literature, but no coherent presentation was available. The present book closes this gap. It includes several pearls from the theory of convex cones, which should be better known.

Fundamentals of Convex Analysis

Author : Jean-Baptiste Hiriart-Urruty
Publisher : Springer Science & Business Media
ISBN 13 : 3642564682
Total Pages : 268 pages
Book Rating : 4.6/5 (425 download)

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Book Synopsis Fundamentals of Convex Analysis by : Jean-Baptiste Hiriart-Urruty

Download or read book Fundamentals of Convex Analysis written by Jean-Baptiste Hiriart-Urruty and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an abridged version of the two volumes "Convex Analysis and Minimization Algorithms I and II" (Grundlehren der mathematischen Wissenschaften Vol. 305 and 306). It presents an introduction to the basic concepts in convex analysis and a study of convex minimization problems (with an emphasis on numerical algorithms). The "backbone" of bot volumes was extracted, some material deleted which was deemed too advanced for an introduction, or too closely attached to numerical algorithms. Some exercises were included and finally the index has been considerably enriched, making it an excellent choice for the purpose of learning and teaching.

Handbook of Convex Geometry

Author : P. M. Gruber
Publisher :
ISBN 13 : 9780444895981
Total Pages : 735 pages
Book Rating : 4.8/5 (959 download)

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Book Synopsis Handbook of Convex Geometry by : P. M. Gruber

Download or read book Handbook of Convex Geometry written by P. M. Gruber and published by . This book was released on 1993 with total page 735 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Convex Geometric Analysis

Author : Keith M. Ball
Publisher : Cambridge University Press
ISBN 13 : 9780521642590
Total Pages : 260 pages
Book Rating : 4.6/5 (425 download)

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Book Synopsis Convex Geometric Analysis by : Keith M. Ball

Download or read book Convex Geometric Analysis written by Keith M. Ball and published by Cambridge University Press. This book was released on 1999-01-28 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: Articles on classical convex geometry, geometric functional analysis, computational geometry, and related areas of harmonic analysis, first published in 1999.

Convex Geometry

Author : Shiri Artstein-Avidan
Publisher : Springer Nature
ISBN 13 : 3031378830
Total Pages : 304 pages
Book Rating : 4.0/5 (313 download)

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Book Synopsis Convex Geometry by : Shiri Artstein-Avidan

Download or read book Convex Geometry written by Shiri Artstein-Avidan and published by Springer Nature. This book was released on 2023-12-13 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects the lecture notes of the Summer School on Convex Geometry, held in Cetraro, Italy, from August 30th to September 3rd, 2021. Convex geometry is a very active area in mathematics with a solid tradition and a promising future. Its main objects of study are convex bodies, that is, compact and convex subsets of n-dimensional Euclidean space. The so-called Brunn--Minkowski theory currently represents the central part of convex geometry. The Summer School provided an introduction to various aspects of convex geometry: The theory of valuations, including its recent developments concerning valuations on function spaces; geometric and analytic inequalities, including those which come from the Lp Brunn--Minkowski theory; geometric and analytic notions of duality, along with their interplay with mass transportation and concentration phenomena; symmetrizations, which provide one of the main tools to many variational problems (not only in convex geometry). Each of these parts is represented by one of the courses given during the Summer School and corresponds to one of the chapters of the present volume. The initial chapter contains some basic notions in convex geometry, which form a common background for the subsequent chapters. The material of this book is essentially self-contained and, like the Summer School, is addressed to PhD and post-doctoral students and to all researchers approaching convex geometry for the first time.

Semidefinite Optimization and Convex Algebraic Geometry

Author : Grigoriy Blekherman
Publisher : SIAM
ISBN 13 : 1611972280
Total Pages : 487 pages
Book Rating : 4.6/5 (119 download)

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Book Synopsis Semidefinite Optimization and Convex Algebraic Geometry by : Grigoriy Blekherman

Download or read book Semidefinite Optimization and Convex Algebraic Geometry written by Grigoriy Blekherman and published by SIAM. This book was released on 2013-03-21 with total page 487 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.

Bodies of Constant Width

Author : Horst Martini
Publisher : Springer
ISBN 13 : 3030038688
Total Pages : 486 pages
Book Rating : 4.0/5 (3 download)

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Book Synopsis Bodies of Constant Width by : Horst Martini

Download or read book Bodies of Constant Width written by Horst Martini and published by Springer. This book was released on 2019-03-16 with total page 486 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first comprehensive monograph to thoroughly investigate constant width bodies, which is a classic area of interest within convex geometry. It examines bodies of constant width from several points of view, and, in doing so, shows surprising connections between various areas of mathematics. Concise explanations and detailed proofs demonstrate the many interesting properties and applications of these bodies. Numerous instructive diagrams are provided throughout to illustrate these concepts. An introduction to convexity theory is first provided, and the basic properties of constant width bodies are then presented. The book then delves into a number of related topics, which include Constant width bodies in convexity (sections and projections, complete and reduced sets, mixed volumes, and further partial fields) Sets of constant width in non-Euclidean geometries (in real Banach spaces, and in hyperbolic, spherical, and further non-Euclidean spaces) The concept of constant width in analysis (using Fourier series, spherical integration, and other related methods) Sets of constant width in differential geometry (using systems of lines and discussing notions like curvature, evolutes, etc.) Bodies of constant width in topology (hyperspaces, transnormal manifolds, fiber bundles, and related topics) The notion of constant width in discrete geometry (referring to geometric inequalities, packings and coverings, etc.) Technical applications, such as film projectors, the square-hole drill, and rotary engines Bodies of Constant Width: An Introduction to Convex Geometry with Applications will be a valuable resource for graduate and advanced undergraduate students studying convex geometry and related fields. Additionally, it will appeal to any mathematicians with a general interest in geometry.

Measures of Symmetry for Convex Sets and Stability

Author : Gabor Toth
Publisher : Springer
ISBN 13 : 3319237330
Total Pages : 278 pages
Book Rating : 4.3/5 (192 download)

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Book Synopsis Measures of Symmetry for Convex Sets and Stability by : Gabor Toth

Download or read book Measures of Symmetry for Convex Sets and Stability written by Gabor Toth and published by Springer. This book was released on 2015-11-26 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook treats two important and related matters in convex geometry: the quantification of symmetry of a convex set—measures of symmetry—and the degree to which convex sets that nearly minimize such measures of symmetry are themselves nearly symmetric—the phenomenon of stability. By gathering the subject’s core ideas and highlights around Grünbaum’s general notion of measure of symmetry, it paints a coherent picture of the subject, and guides the reader from the basics to the state-of-the-art. The exposition takes various paths to results in order to develop the reader’s grasp of the unity of ideas, while interspersed remarks enrich the material with a behind-the-scenes view of corollaries and logical connections, alternative proofs, and allied results from the literature. Numerous illustrations elucidate definitions and key constructions, and over 70 exercises—with hints and references for the more difficult ones—test and sharpen the reader’s comprehension. The presentation includes: a basic course covering foundational notions in convex geometry, the three pillars of the combinatorial theory (the theorems of Carathéodory, Radon, and Helly), critical sets and Minkowski measure, the Minkowski–Radon inequality, and, to illustrate the general theory, a study of convex bodies of constant width; two proofs of F. John’s ellipsoid theorem; a treatment of the stability of Minkowski measure, the Banach–Mazur metric, and Groemer’s stability estimate for the Brunn–Minkowski inequality; important specializations of Grünbaum’s abstract measure of symmetry, such as Winternitz measure, the Rogers–Shepard volume ratio, and Guo’s Lp -Minkowski measure; a construction by the author of a new sequence of measures of symmetry, the kth mean Minkowski measure; and lastly, an intriguing application to the moduli space of certain distinguished maps from a Riemannian homogeneous space to spheres—illustrating the broad mathematical relevance of the book’s subject.

Lectures on Discrete Geometry

Author : Jiri Matousek
Publisher : Springer Science & Business Media
ISBN 13 : 1461300398
Total Pages : 491 pages
Book Rating : 4.4/5 (613 download)

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Book Synopsis Lectures on Discrete Geometry by : Jiri Matousek

Download or read book Lectures on Discrete Geometry written by Jiri Matousek and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 491 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main topics in this introductory text to discrete geometry include basics on convex sets, convex polytopes and hyperplane arrangements, combinatorial complexity of geometric configurations, intersection patterns and transversals of convex sets, geometric Ramsey-type results, and embeddings of finite metric spaces into normed spaces. In each area, the text explains several key results and methods.

Convex Bodies: The Brunn–Minkowski Theory

Author : Rolf Schneider
Publisher : Cambridge University Press
ISBN 13 : 1107601010
Total Pages : 759 pages
Book Rating : 4.1/5 (76 download)

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

Download or read book Convex Bodies: The Brunn–Minkowski Theory written by Rolf Schneider and published by Cambridge University Press. This book was released on 2014 with total page 759 pages. Available in PDF, EPUB and Kindle. Book excerpt: A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research.

Join Geometries

Author : W. Prenowitz
Publisher : Springer Science & Business Media
ISBN 13 : 1461394384
Total Pages : 554 pages
Book Rating : 4.4/5 (613 download)

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Book Synopsis Join Geometries by : W. Prenowitz

Download or read book Join Geometries written by W. Prenowitz and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 554 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main object of this book is to reorient and revitalize classical geometry in a way that will bring it closer to the mainstream of contemporary mathematics. The postulational basis of the subject will be radically revised in order to construct a broad-scale and conceptually unified treatment. The familiar figures of classical geometry-points, segments, lines, planes, triangles, circles, and so on-stem from problems in the physical world and seem to be conceptually unrelated. However, a natural setting for their study is provided by the concept of convex set, which is compara tively new in the history of geometrical ideas. The familiarfigures can then appear as convex sets, boundaries of convex sets, or finite unions of convex sets. Moreover, two basic types of figure in linear geometry are special cases of convex set: linear space (point, line, and plane) and halfspace (ray, halfplane, and halfspace). Therefore we choose convex set to be the central type of figure in our treatment of geometry. How can the wealth of geometric knowledge be organized around this idea? By defini tion, a set is convex if it contains the segment joining each pair of its points; that is, if it is closed under the operation of joining two points to form a segment. But this is precisely the basic operation in Euclid.

Fourier Analysis in Convex Geometry

Author : Alexander Koldobsky
Publisher : American Mathematical Soc.
ISBN 13 : 1470419521
Total Pages : 178 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Fourier Analysis in Convex Geometry by : Alexander Koldobsky

Download or read book Fourier Analysis in Convex Geometry written by Alexander Koldobsky and published by American Mathematical Soc.. This book was released on 2014-11-12 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.