Gruber P. Convex and Discrete Geometry

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440 Geometry of Numbers How seriously Hilbert [501] took the packing problem can be seen from the following question in his 18th problem: How can one arrange most densely in space an infinite number of equal solids of given form, e.g. spheres with given radii or regular tetrahedra with given edges (or in prescribed position), that is, how can one so fit them together that the ratio of the filled to the unfilled space may be as great as possible? In this section we consider lattice packings and packings of translates of a given convex body. After some definitions and simple remarks, we show that densest lattice packings of convex bodies always exist. Then bounds for the number of neighbours in lattice packings of convex bodies are given. Next, we mention an algorithm of Betke and Henk, which permits us to determine the densest lattice packing of convex polytopes in E 3 , and present a lower bound for the maximum density of lattice packings as a consequence of the Minkowski–Hlawka theorem. Finally, we consider, in the planar case, the relation between the lattice and the non-lattice case. For more detailed expositions, see the books of Fejes Tóth [329,330] and Rogers [851], and the surveys of Fejes Tóth [322], Fejes Tóth and Kuperberg [325], Gruber [438] and Bambah [56]. 30.1 Definitions and the Existence of Densest Lattice Packings This section contains the definitions of packing and density and some simple, yet important, properties of packings. In particular, we study the relation between packings of convex bodies and packings of their central symmetrizations, and show the existence of lattice packings of convex bodies of maximum density. Packing of Convex Bodies and the Notion of Density A family of convex bodies in E d is a packing if any two distinct bodies have disjoint interior. We will consider packings of translates and lattice packings of a given proper convex body C, i.e. packings of the form {C + t : t ∈ T } and {C + l : l ∈ L}, where T a discrete set and L a lattice in E d , respectively. If {C +l : l ∈ L} is a lattice packing of C then L is called a packing lattice of C.LetK be the cube {x :|xi| ≤1}. If T is a discrete set in E d then its upper and lower density are lim sup τ→+∞ #(T ∩ τ K ) (2τ) d , lim inf τ→+∞ #(T ∩ τ K ) (2τ) d . If the upper and the lower density of T coincide, their common value is the density δ(T ) of the discrete set T . If the density of T exists, it may be interpreted as the number of points of T per unit volume or, roughly speaking, as the number of points in T divided by the volume of E d .
30 Packing of Convex Bodies 441 Next, given a convex body C and a discrete set T , consider the family {C + t : t ∈ T } of translates of C by the vectors of T .Itsupper and lower density are lim sup τ→+∞ 1 (2τ) d � V � (C + t) ∩ τ K � 1 , lim inf τ→+∞ (2τ) d � V � (C + t) ∩ τ K � . t∈T If they coincide, their common value is called the density δ(C, T ) of the given family. In other sources different definitions are used. For packings and coverings with translates of a convex body, these amount to the same values for the upper and lower densities. For more general families of convex bodies our definitions are still close to the intuitive notion of density and avoid strange occurrences such as packings with upper density +∞. Roughly speaking, the density of a family {C + t : t ∈ T } of translates of C by the vectors of T is the total volume of the bodies divided by the total volume of E d . In the case where this family is a packing, the density may be considered as the proportion of E d which is covered by the bodies of the packing, or as the probability that a ‘random point’ of E d is contained in one of the bodies of the packing. Given a convex body C, letδT (C) and δL(C) denote the supremum of the upper densities of all packings of translates of C, and all lattice packings of C, respectively. δT (C) and δL(C) are called the (maximum) translative packing density and the (maximum) lattice packing density of C, respectively. Clearly, 0 0 such that: (1) F ⊆ σ K . For the proof of the proposition, it is sufficient to show that (2) 2d (τ − σ) d d(L) ≤ #(L ∩ τ K ) ≤ 2d (τ + σ) d d(L) t∈T for τ>σ. The parallelotopes {F + l : l ∈ L} are pairwise disjoint and cover E d . Thus the parallelotopes in this family which intersect τ K ,infact,coverτ K . By (1) these parallelotopes are all contained in (τ + σ)K . Thus if m is their number, we have #(L ∩ τ K ) ≤ m = mV(F) V (F) ≤ V � (τ + σ)K � V (F) = 2d (τ + σ) d . d(L)
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Grundlehren der mathematischen Wiss
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Peter M. Gruber Institute of Discre
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VI Preface theory of finite groups
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Contents Preface ..................
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Contents XI 12 Special Convex Bodie
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Contents XIII 29 Packing of Balls a
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2 Convex Functions 1 Convex Functio
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4 Convex Functions x epi f f C y Fi
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6 Convex Functions Together these i
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8 Convex Functions Theorem 1.4. Let
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10 Convex Functions z = f (x) + f
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12 Convex Functions 1.3 Convexity C
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14 Convex Functions Proof (by affin
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16 Convex Functions Minkowski’s I
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18 Convex Functions (ii) Let x > 0.
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20 Convex Functions 2 Convex Functi
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22 Convex Functions | f (z) − f (
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24 Convex Functions First-Order Dif
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26 Convex Functions Remark. More pr
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28 Convex Functions even more is tr
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30 Convex Functions G(x). The proof
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32 Convex Functions by (15), unifor
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34 Convex Functions 2.4 A Stone-Wei
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36 Convex Functions x x Fig. 2.1. S
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38 Convex Functions F is convex. Us
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40 Convex Bodies results, character
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42 Convex Bodies Convex Hulls Given
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44 Convex Bodies The next result is
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46 Convex Bodies C ∩ L ⊥ clearl
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48 Convex Bodies Suppose not. Then,
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50 Convex Bodies The Centrepoint of
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52 Convex Bodies Since G is open, w
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54 Convex Bodies in one of the two
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56 Convex Bodies or geometric infor
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58 Convex Bodies v HC (u, h(u)) (v,
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60 Convex Bodies C D Fig. 4.3. Sepa
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62 Convex Bodies For the proof of t
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64 Convex Bodies in contradiction t
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66 Convex Bodies (6) R(t) ⊆ E d i
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68 Convex Bodies 5 The Boundary of
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70 Convex Bodies for all y ∈ C an
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72 Convex Bodies the existence of s
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74 Convex Bodies where r(·) is a s
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76 Convex Bodies Hence x ∈ conv e
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78 Convex Bodies N = � � X, O Y
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80 Convex Bodies Further topics of
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82 Convex Bodies We are now ready t
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84 Convex Bodies similarly, D j is
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86 Convex Bodies For the proof of t
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88 Convex Bodies The definition of
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90 Convex Bodies If ei ∈ Pi is no
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92 Convex Bodies Let v(·) denote (
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94 Convex Bodies The present sectio
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96 Convex Bodies Proposition 6.5. L
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98 Convex Bodies Thus Second, Minko
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100 Convex Bodies The Valuation Pro
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102 Convex Bodies Minkowski’s the
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104 Convex Bodies To remedy the sit
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106 Convex Bodies Proof. Applying S
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108 Convex Bodies Third, let C be a
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110 Convex Bodies A Problem of Sant
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112 Convex Bodies sets of S. Ifφ c
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114 Convex Bodies = � (−1) |J|
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116 Convex Bodies If P has dimensio
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118 Convex Bodies Since, by Volland
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120 Convex Bodies that is, V is sim
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122 Convex Bodies Let C, D ∈ C su
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124 Convex Bodies P Q . . . . Fig.
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126 Convex Bodies 7.3 Characterizat
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128 Convex Bodies and the Bi have p
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130 Convex Bodies Since the Riemann
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132 Convex Bodies Seventh, o v3 v2
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134 Convex Bodies Mean Width and th
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136 Convex Bodies � χ(C ∩ mD)
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138 Convex Bodies In the following,
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140 Convex Bodies 7.5 Hadwiger’s
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142 Convex Bodies 8.1 The Classical
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144 Convex Bodies C ∩ H(tC ) C D
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146 Convex Bodies Theorem 8.4. Let
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148 Convex Bodies In the following
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150 Convex Bodies Integration impli
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152 Convex Bodies As a consequence
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154 Convex Bodies Proof. � 1 2 w(
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156 Convex Bodies Let ϱ>0 be the m
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158 Convex Bodies Wulff’s Theorem
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160 Convex Bodies Third, by (8) and
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162 Convex Bodies Since the measure
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164 Convex Bodies An application of
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166 Convex Bodies Since f has Lipsc
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168 Convex Bodies Clearly, the inte
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170 Convex Bodies Note that �st :
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172 Convex Bodies Proposition 9.2.
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174 Convex Bodies Proof. Let ε>0.
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176 Convex Bodies The Brunn-Minkows
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178 Convex Bodies 9.3 Schwarz Symme
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180 Convex Bodies Torsional Rigidit
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182 Convex Bodies The proof of (3)
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184 Convex Bodies � �� 2 2
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188 Convex Bodies Both problems inf
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190 Convex Bodies The Area Measure
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192 Convex Bodies halfspace {z : un
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194 Convex Bodies Let σm be the di
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196 Convex Bodies Thus Minkowski’
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198 Convex Bodies Suppose now that
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200 Convex Bodies vector of St at x
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202 Convex Bodies Outer Billiards F
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204 Convex Bodies For more informat
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206 Convex Bodies with P. Thus, it
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208 Convex Bodies Note that ui ∈
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212 Convex Bodies the maximum of th
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214 Convex Bodies (20) mni ≥ 1 λ
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216 Convex Bodies Prove analogous r
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218 Convex Bodies Heuristic Princip
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220 Convex Bodies λC + x νC + z
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222 Convex Bodies To show that (K +
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224 Convex Bodies The integral here
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226 Convex Bodies In the following
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228 Convex Bodies The first such re
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230 Convex Bodies v o Fig. 12.3. Sy
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232 Convex Bodies What Does the Bou
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234 Convex Bodies Corollary 13.1. L
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236 Convex Bodies Hence ν = 0 and
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238 Convex Bodies geometry, see Sei
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240 Convex Bodies We distinguish th
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242 Convex Bodies by Proposition 13
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244 Convex Polytopes The material i
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246 Convex Polytopes V-Polytopes an
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248 Convex Polytopes Convex cones w
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250 Convex Polytopes Generalized Co
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252 Convex Polytopes d 2 inequaliti
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254 Convex Polytopes The Face Latti
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256 Convex Polytopes Then (6) impli
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258 Convex Polytopes P in F. For an
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260 Convex Polytopes Euler’s Poly
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262 Convex Polytopes bijection betw
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264 Convex Polytopes The Converse o
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266 Convex Polytopes (geometric) fa
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268 Convex Polytopes consists preci
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270 Convex Polytopes The Lower and
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272 Convex Polytopes A problem of S
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274 Convex Polytopes The first cond
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276 Convex Polytopes The existence
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278 Convex Polytopes For d = 3 it f
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280 Convex Polytopes 16 Volume of P
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282 Convex Polytopes by (2). Hence
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284 Convex Polytopes (12) The volum
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286 Convex Polytopes T ∩ Z, respe
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288 Convex Polytopes An Alternative
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290 Convex Polytopes P clearly can
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292 Convex Polytopes 17 Rigidity Ri
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294 Convex Polytopes Fig. 17.2. Cau
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296 Convex Polytopes Call the index
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298 Convex Polytopes of V. Fori ∈
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300 Convex Polytopes Thus (3) impli
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302 Convex Polytopes Proof. (i)⇒(
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304 Convex Polytopes For other pert
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306 Convex Polytopes where λ is a
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308 Convex Polytopes 18.3 The Isope
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310 Convex Polytopes 19 Lattice Pol
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312 Convex Polytopes Embed E d into
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314 Convex Polytopes To see this, n
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316 Convex Polytopes The Coefficien
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318 Convex Polytopes If the triangl
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320 Convex Polytopes � qP(t) = No
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322 Convex Polytopes Now, take into
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324 Convex Polytopes L � P + (n +
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326 Convex Polytopes V (S) = V .Let
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328 Convex Polytopes (7) ψ|H d = 0
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330 Convex Polytopes Theorem 19.6.
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332 Convex Polytopes (15) L(nP) = L
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334 Convex Polytopes have to be sin
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336 Convex Polytopes economics, von
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338 Convex Polytopes Existence of S
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340 Convex Polytopes Description of
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342 Convex Polytopes (6) u piai xq
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344 Convex Polytopes For x ∈ Bd
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346 Convex Polytopes Rational Polyh
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348 Convex Polytopes Totally Dual I
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350 Convex Polytopes of the vectors
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Geometry of Numbers and Aspects of
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Extension to Crystallographic Group
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Relations Between Different Bases 2
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21 Lattices 359 triangle conv{o,(u,
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By definition of bi+1, Thus (6) yie
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21 Lattices 363 A similar, slightly
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21.4 Polar Lattices 21 Lattices 365
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The First Fundamental Theorem 22 Mi
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22 Minkowski’s First Fundamental
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If 22 Minkowski’s First Fundament
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22 Minkowski’s First Fundamental
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Proof. Consider the linear forms l1
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Proof. It is sufficient to consider
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Thus V (C) ≥ V (O) = 2d � �
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Given C and L, the problem arises t
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23 Successive Minima 383 as require
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24 The Minkowski-Hlawka Theorem 24
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(6) (V (J) ≈) � n�= 0 (7) J
Page 399 and 400: 24 The Minkowski-Hlawka Theorem 389
Page 401 and 402: The Variance Theorem of Rogers and
Page 403 and 404: The Selection Theorem of Mahler [68
Page 405 and 406: 26 The Torus Group E d /L 395 where
Page 407 and 408: Proposition 26.2. E d /L is a compa
Page 409 and 410: 26 The Torus Group E d /L 399 theor
Page 411 and 412: 26 The Torus Group E d /L 401 The m
Page 413 and 414: The formula to calculate the Jordan
Page 415 and 416: 27 Special Problems in the Geometry
Page 417 and 418: Asymptotic Estimates 27 Special Pro
Page 419 and 420: 27 Special Problems in the Geometry
Page 421 and 422: 28 Basis Reduction and Polynomial A
Page 423 and 424: (ii) �b1� ≤2 1 2 (d−1) min
Page 425 and 426: 28 Basis Reduction and Polynomial A
Page 427 and 428: and so 28 Basis Reduction and Polyn
Page 429 and 430: Shortest Lattice Vector Problem 28
Page 431 and 432: and therefore �x − m� 2 = �
Page 433 and 434: 29 Packing of Balls and Positive Qu
Page 437 and 438: 29.3 Error Correcting Codes and Bal
Page 449: 30 Packing of Convex Bodies 439 The
Page 453 and 454: Packing and Central Symmetrization
Page 455 and 456: Existence of Densest Lattice Packin
Page 457 and 458: 30 Packing of Convex Bodies 447 Rem
Page 459 and 460: A Lower Estimate for the Maximum La
Page 461 and 462: Lattice Packing and Packing of Tran
Page 463 and 464: a f b o e 2D c Fig. 30.3. Disc and
Page 465 and 466: 31 Covering with Convex Bodies 455
Page 473 and 474: 32 Tiling with Convex Polytopes 463
Page 485 and 486: Define 32 Tiling with Convex Polyto
Page 491 and 492: 33.1 Fejes Tóth’s Inequality for
Page 493 and 494: Noting that 2h 2 v − hhvv = 2π 2
Page 495 and 496: 33 Optimum Quantization 485 The con
Page 497 and 498: (7) � (x, y) : x ∈ K, 0 ≤ y
Page 499 and 500: Next, we prove that (18) lim inf n
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Generalizations 33 Optimum Quantiza
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33 Optimum Quantization 493 set wit
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33 Optimum Quantization 495 In (4)
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33 Optimum Quantization 497 encoder
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34 Koebe’s Representation Theorem
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G 34 Koebe’s Representation Theor
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Di+1 34 Koebe’s Representation Th
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References 1. Achatz, H., Kleinschm
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References 515 36. Arias de Reyna,
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References 517 88. Beer, G., Topolo
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References 519 136. Bohr, H., Molle
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References 521 190. Carathéodory,
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References 523 237. Dantzig, G.B.,
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References 525 286. Edmonds, A.L.,
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References 527 337. Florian, A., Ex
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References 529 385. Goodman, J.E.,
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References 531 435. Gruber, P.M., C
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References 533 487. Heil, E., Über
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References 535 534. Hyuga, T., An e
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References 537 582. Khovanskiĭ (Ho
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References 539 632. Lazutkin, V.F.,
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References 541 684. Mani-Levitska,
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References 543 736. Minkowski, H.,
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References 545 783. Pach, J., Agarw
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References 547 835. Rickert, N.W.,
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References 549 891. Schmidt, E., Di
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References 551 941. Skubenko, B.F.,
Page 563 and 564:
References 553 992. Teissier, B., V
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References 555 1042. Zassenhaus, H.
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558 INDEX Bravais classification of
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560 INDEX existence of elementary v
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562 INDEX Kneser-Macbeath theorem,
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564 INDEX rational lattice, 414 rat
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566 INDEX transference theorem of K
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568 AUTHOR INDEX Björner, 265 Blas
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570 AUTHOR INDEX Giles, J., 1 Giles
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572 AUTHOR INDEX Kuczma, 13, 17 Kü
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574 AUTHOR INDEX Pick, 316 Pisier,
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576 AUTHOR INDEX Tanemura, 464 Tarj
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578 List of Symbols sch, schL Schwa
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289. Manin: Gauge Field Theory and
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