Luisa Handbook of Convex Geometry

1. Auflage 2014
ISBN: 978-0-08-093440-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 765 Seiten, Web PDF

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.

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;Part 3:
Discrete Aspects of Convexity;14
7.1;CHAPTER 3.1.
Geometry of Numbers;16
7.1.1;1. Introduction;18
7.1.2;2. Lattices and the space of lattices;18
7.1.3;3. The fundamental theorems;21
7.1.4;4. The Minkowski–Hlawka theorem;25
7.1.5;5. Successive minima;27
7.1.6;6. Reduction theory;28
7.1.7;7. Selected special problems and results;31
7.1.8;Acknowledgements;34
7.1.9;References;34
7.2;CHAPTER 3.2.
Lattice Points;42
7.2.1;1. Preliminaries;44
7.2.2;2. Centrally symmetric convex bodies;46
7.2.3;3. General convex bodies;48
7.2.4;4. Lattice polytopes;55
7.2.5;5. Lattice polyhedra in combinatorial optimization;59
7.2.6;6. Computational complexity of lattice point problems;63
7.2.7;Acknowledgment;67
7.2.8;References;67
7.3;CHAPTER 3.3.
Packing and Covering with Convex Sets;76
7.3.1;Overview;78
7.3.2;1. Introduction;78
7.3.3;2. Density bounds in d-dimensional Euclidean space;80
7.3.4;3. Packing in non-Euclidean spaces;101
7.3.5;4. Problems in the Euclidean plane;111
7.3.6;5. Additional topics;120
7.3.7;References;127
7.4;CHAPTER 3.4.
Finite Packing and Covering;138
7.4.1;1. Preliminaries;140
7.4.2;2. Sausage problems;142
7.4.3;3. Bin packing;156
7.4.4;4. Miscellaneous packing and covering problems;162
7.4.5;Acknowledgment;168
7.4.6;References;168
7.5;CHAPTER 3.5.
Tilings;176
7.5.1;1· Introduction;178
7.5.2;2. Basic notions;179
7.5.3;3. Plane tilings;183
7.5.4;4. Monohedral tilings;192
7.5.5;5. Non-periodic tilings;198
7.5.6;References;203
7.6;CHAPTER 3.6.
Valuations and Dissections;210
7.6.1;Introduction;212
7.6.2;1. The basic theory;212
7.6.3;2. The classical examples;216
7.6.4;3. The polytope algebra;221
7.6.5;4. Srniple valuations and dissections;232
7.6.6;5. Characterization theorems;245
7.6.7;References;257
7.7;CHAPTER 3.7.
Geometric Crystallography;266
7.7.1;Introduction;268
7.7.2;1. Regular systems of points;268
7.7.3;2. Dirichlet domains;273
7.7.4;3. Translation lattices;279
7.7.5;4. Reduction of quadratic forms;286
7.7.6;5. Finite groups of symmetry operations;292
7.7.7;6. Infinite groups of symmetry operations;300
7.7.8;7. Non-regular systems of points;308
7.7.9;References;310
8;Part 4:
Analytic Aspects of Convexity;320
8.1;CHAPTER 4.1.
Convexity and Differential Geometry;322
8.1.1;Introduction;324
8.1.2;1.
Differential geometric characterization of convexity;325
8.1.3;2. Elementary symmetric functions of principal curvatures respectively principal radii of curvature at Euler points;330
8.1.4;3. Mixed discriminants and mixed volumes;336
8.1.5;4. Differential geometric proof of the Aleksandrov–Fenchel–Jessen inequalities;342
8.1.6;5. Uniqueness theorems for convex hypersurfaces;347
8.1.7;6. Convexity and relative geometry;352
8.1.8;7. Convexity and affine differential geometry;354
8.1.9;References;355
8.2;CHAPTER 4.2.
Convex Functions;358
8.2.1;1. Basic notions;361
8.2.2;2. Differentiability;367
8.2.3;3. Inequalities;377
8.2.4;References;380
8.3;CHAPTER 4.3.
Convexity and Calculus of Variations;382
8.3.1;Introduction;384
8.3.2;1. Extremum problems for functions;385
8.3.3;2. The basic problem of the calculus of variations;386
8.3.4;3. Multiple integrals in the calculus of variations;399
8.3.5;References;406
8.4;CHAPTER 4.4.
On Isoperimetric Theorems of Mathematical Physics;408
8.4.1;1. Introduction;410
8.4.2;2. Historical and bibliographical comments;410
8.4.3;3. Rearrangements;411
8.4.4;4. Capacity;414
8.4.5;5. Torsional rigidity;416
8.4.6;6. Clamped membranes;418
8.4.7;7. Clamped plates;420
8.4.8;References;422
8.5;CHAPTER 4.5. The Local Theory of Normed Spaces and its Applications to Convexity;426
8.5.1;1. Introduction;428
8.5.2;2. Basic concepts;429
8.5.3;3· lnp Subspaces of Banach spaces;442
8.5.4;4. Ellipsoids in local theory;452
8.5.5;5. Distances and projections;462
8.5.6;6. Applications to classical convexity theory in Rn;477
8.5.7;References;488
8.6;CHAPTER 4.6. Nonexpansive Maps and Fixed Points;498
8.6.1;1. Introduction;500
8.6.2;2. Some examples;501
8.6.3;3. Some results (and some history);502
8.6.4;4. Some generalizations;505
8.6.5;5. A few miscellaneous results;508
8.6.6;6. Some other general facts;509
8.6.7;Acknowledgement;510
8.6.8;References;511
8.7;CHAPTER 4.7.
Critical Exponents;514
8.7.1;1. Motivation;516
8.7.2;2. History;520
8.7.3;3. Hilbert space;521
8.7.4;4· Polytopes;525
8.7.5;5. Open problems;530
8.7.6;References;532
8.8;CHAPTER 4.8.
Fourier Series and Spherical Harmonics in Convexity;536
8.8.1;1. Notations and basic concepts;538
8.8.2;2. Geometrie applications of Fourier series;545
8.8.3;3. Geometric applications of spherical harmonics;554
8.8.4;Acknowledgements;567
8.8.5;References;567
8.9;CHAPTER 4.9.
Zonoids and Generalisations;574
8.9.1;1. Introduction;576
8.9.2;2. Basic definitions and properties;576
8.9.3;3. Analytic characterisations of zonoids;582
8.9.4;4. Centrally symmetric bodies and the spherical Radon transform;584
8.9.5;5. Projections onto hyperplanes;589
8.9.6;6. Projection functions on higher rank Grassmannians;592
8.9.7;7. Classes of centrally symmetric bodies;594
8.9.8;8. Zonoids in integral and stochastic geometry;596
8.9.9;References;598
8.10;CHAPTER 4.10.
Baire Categories in Convexity;604
8.10.1;1. Introduction and basic definitions;606
8.10.2;2. A typical proof of a Baire category type result in convexity;606
8.10.3;3. Boundary properties of arbitrary convex bodies;607
8.10.4;4. Smoothness and strict convexity;608
8.10.5;5. Geodesies;610
8.10.6;6. Billiards;611
8.10.7;7. Normals, mirrors and diameters;611
8.10.8;8. Approximation of convex bodies by polytopes;612
8.10.9;9. Points of contact;614
8.10.10;10. Shadow boundaries;614
8.10.11;11. Metric projections;615
8.10.12;12. Miscellaneous results for typical convex bodies;616
8.10.13;13. Starbodies, starsets and compact sets;617
8.10.14;Acknowledgement;618
8.10.15;References;618
9;Part 5:
Stochastic Aspects of Convexity;624
9.1;CHAPTER 5.1.
Integral Geometry;626
9.1.1;1. Preliminaries: Spaces, groups, and measures;628
9.1.2;2. Intersection formulae;630
9.1.3;3. Minkowski addition and projections;637
9.1.4;4. Distance integrals and contact measures;642
9.1.5;5. Extension to the convex ring;648
9.1.6;6. Translative integral geometry and auxiliary zonoids;652
9.1.7;7. Lines and flats through convex bodies;656
9.1.8;References;663
9.2;CHAPTER 5.2.
Stochastic Geometry;668
9.2.1;Preliminaries;670
9.2.2;1. Random points in a convex body;672
9.2.3;2. Random flats intersecting a convex body;679
9.2.4;3. Random convex bodies;681
9.2.5;4. Random sets;684
9.2.6;5. Point processes;687
9.2.7;6. Random surfaces;696
9.2.8;7. Random mosaics;700
9.2.9;8. Stereology;705
9.2.10;References;708
10;Author Index;716
11;Subject Index;750