Volumetric Discrete Geometry

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Authors
Symbols
I Selected Topics
1 Volumetric Properties of (m, d)-scribed Polytopes
1.1 The isoperimetric inequality
1.2 Discrete isoperimetric inequalities: volume of polytopes circumscribed about a sphere
1.3 Volume of polytopes inscribed in a sphere
1.4 Polyhedra midscribed to a sphere
1.5 Research Exercises
2 Volume of the Convex Hull of a Pair of Convex Bodies
2.1 Volume of the convex hull of a pair of convex bodies in Euclidean space
2.2 Volume of the convex hull of a pair of convex bodies in normed spaces
2.3 Research Exercises
3 The Kneser-Poulsen Conjecture Revisited
3.1 The Kneser-Poulsen conjecture
3.2 The Kneser-Poulsen conjecture for continuous contractions of unions and intersections of balls
3.3 The Kneser-Poulsen conjecture for contractions of unions and intersections of disks in E2
3.4 The Kneser-Poulsen conjecture for uniform contractions of r-ball polyhedra in Ed, Sd and ℍd
3.5 The Kneser-Poulsen conjecture for contractions of unions and intersections of disks in S2 and ℍ2
3.6 Research Exercises
4 Volumetric Bounds for Contact Numbers
4.1 Description of the basic geometric questions
4.2 Motivation from materials science
4.3 Largest contact numbers for congruent circle packings
4.3.1 The Euclidean plane
4.3.2 Spherical and hyperbolic planes
4.4 Largest contact numbers for unit ball packings in E3
4.5 Upper bounding the contact numbers for packings by translates of a convex body in Ed
4.6 Contact numbers for digital and totally separable packings of unit balls in Ed
4.7 Bounds for contact numbers of totally separable packings by translates of a convex body in Ed with d = 1, 2, 3, 4
4.7.1 Separable Hadwiger numbers
4.7.2 One-sided separable Hadwiger numbers
4.7.3 Maximum separable contact numbers
4.8 Appendix: Hadwiger numbers of topological disks
4.9 Research Exercises
5 More on Volumetric Properties of Separable Packings
5.1 Solution of the contact number problem for smooth strictly convex domains in E2
5.2 The separable Oler’s inequality and its applications in E2
5.2.1 Oler’s inequality
5.2.2 An analogue of Oler’s inequality for totally separable translative packings
5.2.3 On the densest totally separable translative packings
5.2.4 On the smallest area convex hull of totally separable translative finite packings
5.3 Higher dimensional results: minimizing the mean projections of finite ρ-separable packings in Ed
5.4 Research Exercises
II Selected Proofs
6 Proofs on Volumetric Properties of (m, d)-scribed Polytopes
6.1 Proof of Theorem 3
6.2 Proofs of Theorems 10 and 11
6.3 Proof of Theorem 14
6.3.1 Preliminaries
6.3.2 Proof of Theorem 14 for n ≤ 6
6.3.3 Proof of Theorem 14 for n = 7
6.3.4 Proof of Theorem 14 for n = 8
6.4 Proofs of Theorems 16, 17 and 18
6.4.1 Proof of Theorem 16 and some lemmas for Theorems 17 and 18
6.4.2 Proofs of Theorems 17 and 18
6.5 Proof of Theorem 21
6.6 Proof of Theorem 22
6.7 Proof of Theorem 27
6.8 Proofs of Theorems 28 and 29
6.8.1 Preliminaries and the main idea of the proofs
6.8.2 The main lemma of the proofs
6.8.3 Proof of Theorem 28
6.8.4 Proof of Theorem 29
7 Proofs on the Volume of the Convex Hull of a Pair of Convex Bodies
7.1 Proofs of Theorems 32 and 33
7.1.1 Proof of Theorem 32
7.1.2 Proof of Theorem 33
7.2 Proofs of Theorems 34, 36, 37 and 40
7.2.1 Preliminaries
7.2.2 Proofs of the Theorems
7.3 Proofs of Theorems 41 and 46
7.3.1 Proof of Theorem 41
7.3.2 Proof of Theorem 46
7.4 Proof of Theorem 53
7.5 Proof of Theorem 54
7.6 Proofs of Theorems 57 and 58
7.6.1 Proof of Theorem 57
7.6.2 Proof of Theorem 58
7.7 Proofs of Theorems 59 and 60
7.7.1 The proof of the left-hand side inequality in (ii)
7.7.2 The proof of the right-hand side inequality in (ii)
7.7.3 The proofs of (i), (iii) and (iv)
8 Proofs on the Kneser-Poulsen Conjecture
8.1 Proof of Theorem 67
8.2 Proof of Theorem 68
8.3 Proof of Theorem 69
8.4 Proof of Theorem 72
8.5 Proof of Theorem 73
8.5.1 Proof of (i) in Theorem 73
8.5.2 Proof of (ii) in Theorem 73
8.5.3 Proof of (iii) in Theorem 73
8.6 Proof of Theorem 74
8.6.1 The spherical leapfrog lemma
8.6.2 Smooth contractions via Schläfli’s differential formula
8.6.3 From higher- to lower-dimensional spherical volume
8.6.4 Putting pieces together
8.7 Proof of Theorem 75
8.8 Proof of Theorem 76
8.8.1 Proof of Part (i) of Theorem 76
8.8.2 Proof of Part (ii) of Theorem 76
8.9 Proof of Theorem 78
8.10 Proof of Theorem 79
8.10.1 Basic results on central sets of ball-polytopes
8.10.2 An extension theorem via piecewise isometries
8.10.3 Deriving Theorem 79 from the preliminary results
9 Proofs on Volumetric Bounds for Contact Numbers
9.1 Proof of Theorem 87
9.1.1 An upper bound for sphere packings: Proof of (i)
9.1.2 An upper bound for the fcc lattice: Proof of (ii)
9.1.3 Octahedral unit sphere packings: Proof of (iii)
9.2 Proof of Theorem 88
9.3 Proof of Theorem 92
9.4 Proof of Theorem 93
9.5 Proof of Theorem 94
9.6 Proof of Theorem 95
9.7 Proof of Theorem 98
9.8 Proof of Theorem 99
9.9 Proof of Theorem 100
9.10 Proofs of Theorems 101, 102, and 103
9.10.1 Linearization, fundamental properties
9.10.2 Proofs of Theorems 101 and 102
9.10.3 Proof of Theorem 103
9.10.4 Remarks
9.11 Proof of Theorem 108
9.12 Proof of Theorem 110
9.13 Proof of Theorem 111
10 More Proofs on Volumetric Properties of Separable Packings
10.1 Proof of Theorem 113
10.2 Proof of Theorem 116
10.3 Proof of Corollary 117
10.4 Proof of Theorem 119
10.5 Proof of Theorem 123
10.6 Proof of Theorem 125
10.7 Proof of Theorem 126
10.8 Proof of Theorem 130
11 Open Problems: An Overview
11.1 Chapter 1
11.2 Chapter 2
11.3 Chapter 3
11.4 Chapter 4
11.5 Chapter 5
Bibliography
Index