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Author: I. E. Leonard
Publisher: John Wiley & Sons
ISBN: 1119022665
Category : Mathematics
Languages : en
Pages : 340
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Book Description
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.
Author: I. E. Leonard
Publisher: John Wiley & Sons
ISBN: 1119022665
Category : Mathematics
Languages : en
Pages : 340
Get Book Here
Book Description
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.
Author: Steven R. Lay
Publisher: Courier Corporation
ISBN: 0486458032
Category : Mathematics
Languages : en
Pages : 260
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Book Description
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.
Author: Bozzano G Luisa
Publisher: Elsevier
ISBN: 0080934404
Category : Mathematics
Languages : en
Pages : 769
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Book Description
Handbook of Convex Geometry, Volume B offers a survey of convex geometry and its many ramifications and connections with other fields of mathematics, including convexity, lattices, crystallography, and convex functions. The selection first offers information on the geometry of numbers, lattice points, and packing and covering with convex sets. Discussions focus on packing in non-Euclidean spaces, problems in the Euclidean plane, general convex bodies, computational complexity of lattice point problem, centrally symmetric convex bodies, reduction theory, and lattices and the space of lattices. The text then examines finite packing and covering and tilings, including plane tilings, monohedral tilings, bin packing, and sausage problems. The manuscript takes a look at valuations and dissections, geometric crystallography, convexity and differential geometry, and convex functions. Topics include differentiability, inequalities, uniqueness theorems for convex hypersurfaces, mixed discriminants and mixed volumes, differential geometric characterization of convexity, reduction of quadratic forms, and finite groups of symmetry operations. The selection is a dependable source of data for mathematicians and researchers interested in convex geometry.
Author: W. Prenowitz
Publisher: Springer Science & Business Media
ISBN: 1461394384
Category : Mathematics
Languages : en
Pages : 554
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Book Description
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.
Author: Grigoriy Blekherman
Publisher: SIAM
ISBN: 1611972280
Category : Mathematics
Languages : en
Pages : 487
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Book Description
An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.
Author: Jiri Matousek
Publisher: Springer Science & Business Media
ISBN: 1461300398
Category : Mathematics
Languages : en
Pages : 491
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Book Description
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.
Author: Valeriu Soltan
Publisher: World Scientific
ISBN: 9811202133
Category : Mathematics
Languages : en
Pages : 611
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Book Description
The book provides a self-contained and systematic treatment of algebraic and topological properties of convex sets in the n-dimensional Euclidean space. It benefits advanced undergraduate and graduate students with various majors in mathematics, optimization, and operations research. It may be adapted as a primary book or an additional text for any course in convex geometry or convex analysis, aimed at non-geometers. It can be a source for independent study and a reference book for researchers in academia.The second edition essentially extends and revises the original book. Every chapter is rewritten, with many new theorems, examples, problems, and bibliographical references included. It contains three new chapters and 100 additional problems with solutions.
Author: Tadao Oda
Publisher: Springer
ISBN: 9783642725494
Category : Mathematics
Languages : en
Pages : 0
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Book Description
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.
Author: Guillaume Aubrun
Publisher: American Mathematical Soc.
ISBN: 1470434687
Category : Mathematics
Languages : en
Pages : 442
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Book Description
The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twenty-first century, and quantum information theory (QIT) provides the mathematical framework for that quest. Over the last dozen or so years, it has become clear that quantum information theory is closely linked to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability), a field also known as asymptotic geometric analysis (AGA). In a nutshell, asymptotic geometric analysis investigates quantitative properties of convex sets, or other geometric structures, and their approximate symmetries as the dimension becomes large. This makes it especially relevant to quantum theory, where systems consisting of just a few particles naturally lead to models whose dimension is in the thousands, or even in the billions. Alice and Bob Meet Banach is aimed at multiple audiences connected through their interest in the interface of QIT and AGA: at quantum information researchers who want to learn AGA or apply its tools; at mathematicians interested in learning QIT, or at least the part of QIT that is relevant to functional analysis/convex geometry/random matrix theory and related areas; and at beginning researchers in either field. Moreover, this user-friendly book contains numerous tables and explicit estimates, with reasonable constants when possible, which make it a useful reference even for established mathematicians generally familiar with the subject.
Author: Stephen P. Boyd
Publisher: Cambridge University Press
ISBN: 9780521833783
Category : Business & Economics
Languages : en
Pages : 744
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Book Description
Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.
