Measures Of Symmetry For Convex Sets And Stability
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Measures Of Symmetry For Convex Sets And Stability
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Author : Gabor Toth
language : en
Publisher:
Release Date : 2015
Measures Of Symmetry For Convex Sets And Stability written by Gabor Toth and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with categories.
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.
Measures Of Symmetry For Convex Sets And Stability
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Author : Gabor Toth
language : en
Publisher: Springer
Release Date : 2015-11-26
Measures Of Symmetry For Convex Sets And Stability written by Gabor Toth and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-11-26 with Mathematics categories.
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.
Similarity And Symmetry Measures For Convex Sets Based On Minkowski Addition
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Author : Hendricus Johannes Adrianus Maria Heijmans
language : en
Publisher:
Release Date : 1996
Similarity And Symmetry Measures For Convex Sets Based On Minkowski Addition written by Hendricus Johannes Adrianus Maria Heijmans and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with categories.
Symmetry Points Of A Convex Set
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Author : Alexandre Belloni
language : en
Publisher:
Release Date : 2006
Symmetry Points Of A Convex Set written by Alexandre Belloni and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with categories.
Given a convex body S and a point x in S, let sym(x,S) denote the symmetry value of x in S: sym(x,S):= max{t : x + t(x - y) is in S for every y in S}, which essentially measures how symmetric S is about the point x, and define sym(S):=max{sym(x,S) : x in S}. We call x* a symmetry point of S if x* achieves the above supremum. These symmetry measures are all invariant under invertible affine transformation and/or change in norm, and so are of interest in the study of the geometry of convex sets. In this study we demonstrate various properties of sym(x,S), including relations with convex geometry quantities like volume, distance and diameter, and cross-ratio distance. When S is polyhedral of the form {x : Ax
Selected Topics In Convex Geometry
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Author : Maria Moszynska
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-10-03
Selected Topics In Convex Geometry written by Maria Moszynska and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-10-03 with Mathematics categories.
Examines in detail those topics in convex geometry that are concerned with Euclidean space Enriched by numerous examples, illustrations, and exercises, with a good bibliography and index Requires only a basic knowledge of geometry, linear algebra, analysis, topology, and measure theory Can be used for graduates courses or seminars in convex geometry, geometric and convex combinatorics, and convex analysis and optimization
Lectures On Convex Geometry
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Author : Daniel Hug
language : en
Publisher: Springer Nature
Release Date : 2020-08-27
Lectures On Convex Geometry written by Daniel Hug and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-08-27 with Mathematics categories.
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.
Convex Bodies The Brunn Minkowski Theory
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Author : Rolf Schneider
language : en
Publisher: Cambridge University Press
Release Date : 2014
Convex Bodies The Brunn Minkowski Theory written by Rolf Schneider and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with Mathematics categories.
A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research.
Handbook Of Convex Geometry
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Author : Bozzano G Luisa
language : en
Publisher: North Holland
Release Date : 1993-08-24
Handbook Of Convex Geometry written by Bozzano G Luisa and has been published by North Holland this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-08-24 with Mathematics categories.
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.
Handbook Of Convex Geometry
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Author :
language : en
Publisher: North Holland
Release Date : 1993-09-07
Handbook Of Convex Geometry written by and has been published by North Holland this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-09-07 with Mathematics categories.
Handbook of Convex Geometry, Volume A offers a survey of convex geometry and its many ramifications and relations with other areas of mathematics, including convexity, geometric inequalities, and convex sets. The selection first offers information on the history of convexity, characterizations of convex sets, and mixed volumes. Topics include elementary convexity, equality in the Aleksandrov-Fenchel inequality, mixed surface area measures, characteristic properties of convex sets in analysis and differential geometry, and extensions of the notion of a convex set. The text then reviews the standard isoperimetric theorem and stability of geometric inequalities. The manuscript takes a look at selected affine isoperimetric inequalities, extremum problems for convex discs and polyhedra, and rigidity. Discussions focus on include infinitesimal and static rigidity related to surfaces, isoperimetric problem for convex polyhedral, bounds for the volume of a convex polyhedron, curvature image inequality, Busemann intersection inequality and its relatives, and Petty projection inequality. The book then tackles geometric algorithms, convexity and discrete optimization, mathematical programming and convex geometry, and the combinatorial aspects of convex polytopes. The selection is a valuable source of data for mathematicians and researchers interested in convex geometry.
Elements Of Mathematics
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Author : Gabor Toth
language : en
Publisher: Springer Nature
Release Date : 2021-09-23
Elements Of Mathematics written by Gabor Toth and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-09-23 with Mathematics categories.
This textbook offers a rigorous presentation of mathematics before the advent of calculus. Fundamental concepts in algebra, geometry, and number theory are developed from the foundations of set theory along an elementary, inquiry-driven path. Thought-provoking examples and challenging problems inspired by mathematical contests motivate the theory, while frequent historical asides reveal the story of how the ideas were originally developed. Beginning with a thorough treatment of the natural numbers via Peano’s axioms, the opening chapters focus on establishing the natural, integral, rational, and real number systems. Plane geometry is introduced via Birkhoff’s axioms of metric geometry, and chapters on polynomials traverse arithmetical operations, roots, and factoring multivariate expressions. An elementary classification of conics is given, followed by an in-depth study of rational expressions. Exponential, logarithmic, and trigonometric functions complete the picture, driven by inequalities that compare them with polynomial and rational functions. Axioms and limits underpin the treatment throughout, offering not only powerful tools, but insights into non-trivial connections between topics. Elements of Mathematics is ideal for students seeking a deep and engaging mathematical challenge based on elementary tools. Whether enhancing the early undergraduate curriculum for high achievers, or constructing a reflective senior capstone, instructors will find ample material for enquiring mathematics majors. No formal prerequisites are assumed beyond high school algebra, making the book ideal for mathematics circles and competition preparation. Readers who are more advanced in their mathematical studies will appreciate the interleaving of ideas and illuminating historical details.