Finite Packing and Covering

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Cambridge University Press, Aug 2, 2004 - Mathematics - 380 pages
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Finite arrangements of convex bodies were intensively investigated in the second half of the 20th century. Connections to many other subjects were made, including crystallography, the local theory of Banach spaces, and combinatorial optimization. This book, the first one dedicated solely to the subject, provides an in-depth state-of-the-art discussion of the theory of finite packings and coverings by convex bodies. It contains various new results and arguments, besides collecting those scattered around in the literature, and provides a comprehensive treatment of problems whose interplay was not clearly understood before. In order to make the material more accessible, each chapter is essentially independent, and two-dimensional and higher-dimensional arrangements are discussed separately. Arrangements of congruent convex bodies in Euclidean space are discussed, and the density of finite packing and covering by balls in Euclidean, spherical and hyperbolic spaces is considered.
  

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Contents

Congruent Domains in the Euclidean Plane
3
Translative Arrangements
34
Parametric Density
74
Packings of Circular Discs
97
Coverings by Circular Discs
135
Packings and Coverings by Spherical Balls
175
Congruent Convex Bodies
201
Packings and Coverings by Unit Balls
220
Surfaces of Constant Curvature
332
Mixed Areas in X2
340
Polyhedral Sets and Polytopes in R
341
A S Associated Balls and Ellipsoids in Rd
343
Volume Surface Area and Lebesgue Measure in Rd
344
Hausdorff Measure and Lipschitz Functions in Rd
347
Intrinsic Volumes in Rd
348
Mixed Volumes in Rd
351

Translative Arrangements
243
Parametric Density
287
Background
325
The Space of Compact Convex Sets in Rd
330
The Spherical Space and the Hyperbolic Space
331
Lattices in Rd and Tori
353
A Little Bit of Probability
355
Bibliography
357
Index
379
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