## Finite Packing and CoveringFinite 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 |

357 | |

379 | |

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### Common terms and phrases

A(Cn absolute constant affine affine hull angle assume asymptotic balls of radius Betke Boroczky Jr cell complex cell decomposition centrally symmetric centres circular discs circumradius compact convex set conjecture convex body convex domain convex hull Corollary deduce define denote dimensional Dirichlet-Voronoi cell discs of radius distance edge endpoints equality holds Euclidean exists exterior normal Fejes Toth finite packings follows Given a convex Hadwiger number hence homothetic hyperplane inequality intersects Jordan measurable lattice packing least Lemma lower bound maximal number Minkowski inequality nonoverlapping translates nonoverlapping unit o-symmetric convex domain optimal packing orthogonal packing lattice packings and coverings parallelepiped parallelogram parametric density points polygon polytope proof of Theorem Proposition proved QC(K r(Dn respect satisfies sausage segment simplex simplicial complex spherical balls strictly convex tends to infinity tiling triangle bound unit balls unit discs verify vertex vertices write yields Zong