Luisa Handbook of Convex Geometry

1. Auflage 2014
ISBN: 978-0-08-093439-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 801 Seiten, Web PDF

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.

Autoren/Hrsg.

Luisa, Bozzano G

Inhaltsverzeichnis

1;Front Cover;1
2;Handbook of Convex Geometry;4
3;Copyright Page;5
4;Table of Contents;8
5;Preface;6
6;List of Contributors;12
7;CHAPTER 0. History of Convexity;14
7.1;1. Introduction;16
7.2;2. Antiquity;16
7.3;3. Modern times up to the 18th century;17
7.4;4. The 19th century with glimpses into the 20th century;18
7.5;5. From the 19th to the 20th century and the 20th century;20
7.6;Acknowledgements;28
8;Part 1: Classical Convexity;30
8.1;CHAPTER 1.1 Characterizations of Convex Sets;32
8.1.1;1. The classical characterizations;34
8.1.2;2. Characteristic properties of convex sets in analysis and differential geometry;35
8.1.3;3. Topology. Combinatorics;38
8.1.4;4. Extensions of the notion of a convex set. Realizations;42
8.1.5;5. Measures of convexity;45
8.1.6;6. Looking out;46
8.1.7;References;48
8.2;CHAPTER 1.2 Mixed Volumes;56
8.2.1;1. Elementary convexity;58
8.2.2;2. Mixed volumes;59
8.2.3;3. Quermassintegrals and intrinsic volumes;62
8.2.4;4. Mixed surface area measures;65
8.2.5;5. Symmetrization;67
8.2.6;6. Fundamental inequalities;70
8.2.7;7. Equality in the Aleksandrov-Fenchel inequality;73
8.2.8;8. Additional inequalities;77
8.2.9;Acknowledgment;79
8.2.10;References;79
8.3;CHAPTER 1.3 The Standard Isoperimetric Theorem;86
8.3.1;1. Introduction;88
8.3.2;2. The isoperimetric theorem for the Euclidean plane;92
8.3.3;3. The isoperimetric theorem for the Euclidean n -dimensional space;97
8.3.4;References;133
8.4;CHAPTER 1.4 Stability of Geometric Inequalities;138
8.4.1;1. Introduction;140
8.4.2;2. Bonnesen's inequality and its consequences;143
8.4.3;3. The Brunn-Minkowski inequality;146
8.4.4;4. Inequalities for mixed volumes;150
8.4.5;5. Inequalities for the mean projection measures and the isoperimetric inequality;153
8.4.6;6. Cap bodies, bodies of constant width, equichordal sets, packings and coverings;157
8.4.7;7. Projections of convex bodies;159
8.4.8;References;161
8.5;CHAPTER 1.5 Selected Affine Isoperimetric Inequalities;164
8.5.1;Introduction;166
8.5.2;1. Inequalities for random simplices in a convex body;166
8.5.3;2. The Busemann-Petty centroid inequality;168
8.5.4;3. The Petty projection inequality;169
8.5.5;4. Petty's affine projection inequality;170
8.5.6;5. Relationship between the Busemann-Petty centroid and Petty projection inequalities;171
8.5.7;6. Relationship between the Busemann-Petty centroid inequality and Petty's affine projection inequality;172
8.5.8;7. The Busemann intersection inequality and its relatives;173
8.5.9;8. Petty's geominimal surface area inequality and its relatives;176
8.5.10;9. The curvature image inequality;177
8.5.11;10. The affine isoperimetric inequality;177
8.5.12;11. The Blaschke-Santaló inequality;178
8.5.13;12. Open problems;180
8.5.14;Acknowledgements;184
8.5.15;References;184
8.6;CHAPTER 1.6 Extremum Problems for Convex Discs and Polyhedra;190
8.6.1;1. Introduction and notation;192
8.6.2;2. Some extremum problems for convex discs;194
8.6.3;3. Bounds for the volume of a convex polyhedron;210
8.6.4;4. Surface area and edge-curvature;215
8.6.5;5. Total length of the edges of a polyhedron;218
8.6.6;6. The isoperimetric problem for convex polyhedra;221
8.6.7;Acknowledgement;227
8.6.8;References;227
8.7;CHAPTER 1.7 Rigidity;236
8.7.1;1. Introduction;238
8.7.2;2. Early results;239
8.7.3;3. Basic definitions and basic results;245
8.7.4;4. Infinitesimal and static rigidity related to surfaces;253
8.7.5;5. Second-order rigidity and pre-stress stability;269
8.8;CHAPTER 1.8 Convex Surfaces, Curvature and Surface Area Measures;286
8.8.1;Introduction;288
8.8.2;1. First order boundary structure of convex bodies;288
8.8.3;2. Pointwise curvatures;292
8.8.4;3. Curvature measures and surface area measures;294
8.8.5;4. Special cases and relations to local shape;297
8.8.6;5. Minkowski's theorem and other existence problems;300
8.8.7;6. Uniqueness and stability results;305
8.8.8;References;309
8.9;CHAPTER 1.9 The Space of Convex Bodies;314
8.9.1;1. Introduction;316
8.9.2;2. Lattice structure;316
8.9.3;3. Semigroup structure and embedding;317
8.9.4;4. Metrics;320
8.9.5;5. Topology;323
8.9.6;6. Measure and category;324
8.9.7;7. Entropy and generalized dimension;324
8.9.8;Acknowledgements;325
8.9.9;References;325
8.10;CHAPTER 1.10 Aspects of Approximation of Convex Bodies;332
8.10.1;1. Introduction;334
8.10.2;2. Some definitions;334
8.10.3;3. Properties of best approximating polytopes for d = 2;336
8.10.4;4. Upper bounds;337
8.10.5;5. Asymptotic estimates;339
8.10.6;6. Algorithmic and asymptotic step-by-step approximation;345
8.10.7;7. Approximation of special (classes of) convex bodies;347
8.10.8;8. Symmetrization;349
8.10.9;9. Miscellanea;350
8.10.10;Acknowledgements;352
8.10.11;References;352
8.11;CHAPTER 1.11 Special Convex Bodies;360
8.11.1;1. Introduction;362
8.11.2;2. Simplices;368
8.11.3;3. Ellipsoids;371
8.11.4;4. Centrally symmetric convex bodies;374
8.11.5;5. Convex bodies of constant width;376
8.11.6;References;381
9;Part 2: Combinatorial Aspects of Convexity;400
9.1;CHAPTER 2.1 Helly, Radon, and Carathéodory Type Theorems;402
9.1.1;1. Introduction;404
9.1.2;PART I. HELLY'S THEOREM, AND THE COMBINATORIAL GEOMETRY OF FAMILIES OF CONVEX SETS;405
9.1.3;2. Helly's Theorem;405
9.1.4;3. Generalizations of Helly's Theorem;408
9.1.5;4. Spherical and topological Helly-type theorems;413
9.1.6;5. Other Helly-type theorems;417
9.1.7;6. Piercing and coloring properties of convex sets;419
9.1.8;7. Common transversals;423
9.1.9;8. Intersection patterns of convex sets;428
9.1.10;PART II. RADON'S AND CARATHÉODORY'S THEOREM, AND THE CONVEXITY PROPERTIES OF CONFIGURATIONS OF POINTS;434
9.1.11;9. Radon's Theorem and its relatives;434
9.1.12;10. The theorems of Carathéodory and Steinitz;443
9.1.13;11. The theorems of Kirchberger and Krasnosel'skii;446
9.2;CHAPTER 2.2 Problems in Discrete and Combinatorial Geometry;462
9.2.1;Introduction;464
9.2.2;1. Sets of points and related topics;464
9.2.3;2. Finite tilings of sets;474
9.2.4;Acknowledgements;487
9.2.5;References;487
9.3;CHAPTER 2.3 Combinatorial Aspects of Convex Polytopes;498
9.3.1;1. Definitions and fundamental results;500
9.3.2;2. Shellings;501
9.3.3;3. Algebraic methods;509
9.3.4;4. Gale transforms and diagrams;524
9.3.5;5. Graphs of polytopes;533
9.3.6;6. Combinatorial structure;536
9.3.7;References;541
9.4;CHAPTER 2.4 Polyhedral Manifolds;548
9.4.1;1. Preliminaries;550
9.4.2;2. Embeddability;553
9.4.3;3. Minimality properties of polyhedra;555
9.4.4;4. Combinatorially regular polyhedra and related topics;558
9.4.5;References;561
9.5;CHAPTER 2.5 Oriented Matroids;568
9.5.1;1. Introduction;570
9.5.2;2. Models in oriented matroid theory;581
9.5.3;3. Matroid polytopes;588
9.5.4;4. Matroid theory in convexity;598
9.5.5;5. Polytopes versus matroid polytopes;601
9.5.6;References;605
9.6;CHAPTER 2.6 Algebraic Geometry and Convexity;616
9.6.1;1. Introduction and historical notes;618
9.6.2;2. Definition of toric varieties;620
9.6.3;3. Blow ups and stellar subdivisions;625
9.6.4;4. Resolution of singularities;627
9.6.5;5. Invertible sheaves, Picard group, and projective toric varieties;629
9.6.6;6. Homology;632
9.6.7;7. Research problems;634
9.6.8;8. Dictionary;637
9.6.9;References;637
9.7;CHAPTER 2.7 Mathematical Programming and Convex Geometry;640
9.7.1;Introduction;642
9.7.2;1. Terminology for convex optimization;643
9.7.3;2. Some qualitative aspects of convex minimization;645
9.7.4;3. Some qualitative aspects of convex maximization;646
9.7.5;4. General background for linear programming;648
9.7.6;5. Duality; primal-dual algorithm;649
9.7.7;6. Fourier-Motzkin elimination;652
9.7.8;7. The simplex method;653
9.7.9;8. Center of gravity cuts and the ellipsoid method;660
9.7.10;9. Karmarkar's protective algorithm;666
9.7.11;10. Analytic center, path-following methods;672
9.7.12;11. Strongly polynomial algorithms;676
9.7.13;Acknowledgement;677
9.7.14;References;677
9.8;CHAPTER 2.8 Convexity and Discrete Optimization;688
9.8.1;1. Introduction;690
9.8.2;2. Convex, mixed-integer programming;691
9.8.3;3. Feasibility and computational complexity of integer programs;694
9.8.4;4. Integral polyhedra;695
9.8.5;5. Polyhedral combinatorics;697
9.8.6;6. Combinatorial optimization problems arising from convex geometric configurations;705
9.8.7;References;707
9.9;CHAPTER 2.9 Geometric Algorithms;712
9.9.1;1. Introduction;714
9.9.2;2. Topics in computational geometry;716
9.9.3;3. Concluding remarks;742
9.9.4;References;743
10;Author Index;750
11;Subject Index;784