Gruber P. Convex and Discrete Geometry

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450 Geometry of Numbers Let A be the set of all lattices in L(1) which contain a point �= o in the convex body 2C of volume 2 −ψ(d) = o(1). Then Siegel’s mean value theorem shows that µ(A) ≤ � L(1) Thus, we have the following: # ∗ (L ∩ 2C) dµ(L) = V (2C) = o(1). All lattices L ∈ L(1), with a set of exceptions of measure o(1), provide a packing of the convex body C with density 2 −d−ψ(d) . Supposing that, for the Minkowski–Hlawka theorem, there is no essential improvement possible for certain o-symmetric convex bodies, this shows the following: a large majority of lattices of determinant 1 provide lattice packings of such bodies with density close to the maximum lattice packing density. 30.4 Lattice Packing Versus Packing of Translates Roughly speaking, the geometry of numbers deals with regular configurations, in particular with lattice packing, covering and tiling of convex and, possibly, nonconvex bodies. Discrete geometry investigates the irregular case, in particular packing, covering and tiling of translates and congruent copies of convex and non-convex bodies. Classical results in E 2 say that, in several cases, general extremal configurations are no better than the corresponding extremal lattice configurations. An example is a result of Fejes Tóth and Rogers. It says that densest lattice packings of convex discs have maximum density among all packings by translates. For more information see the books of Fejes Tóth [327, 329] and Pach and Agarwal [783]. A stability result of the author [436] gives information on the geometric appearance of general packings of circular discs of maximum density. Such packings are asymptotically regular hexagonal. For other results of this type see Sects. 31.4 and 33.4 and the author’s survey [438]. For d ≥ 3 the only pertinent result for convex bodies is due to Hales. It says that δL(B 3 ) = δT (B 3 ), see Sect. 29.2. (An earlier result of Bezdek and Kuperberg [109] deals with unbounded circular cylinders.) So far there is no example known of a convex body in E d , d ≥ 3, with the property that the maximum lattice packing density is smaller than the maximum density of a packing by translates. If non-convex bodies are admitted, examples for this phenomenon are known, see Szabó [980]. Compare also the discussion in Sect. 32.3. Bezdek and Kuperberg [110] specified, for each d ≥ 3, packings of congruent ellipsoids in E d which have density larger than δL(B d ). In the following we give a proof of the result of Fejes Tóth [328,329] and Rogers [846]. The special case of solid circular discs is due to Thue [996,997] and Fejes Tóth [327]. For a direct proof of the latter, see also Sect. 33.4.
Lattice Packing and Packing of Translates in E 2 We show the following result of Fejes Tóth and Rogers: 30 Packing of Convex Bodies 451 Theorem 30.5. Let C be a proper convex disc in E 2 . Then δT (C) = δL(C). The following proof was given by Fejes Tóth [331]. It seems to be the shortest proof known. Proof. Since the strictly convex discs are dense among all convex discs, it is sufficient to prove this result for strictly convex discs. The result is first proved for triangular discs, that is strictly convex discs C contained in a convex hexagon with vertices a, b, c, d, e, f, parallel opposite edges and such that a, c, e ∈ C. Callthe line through a and b vertical, a and e the left and right and c the bottom vertex of C (Fig. 30.1). The first step in the proof of the theorem is to show the following: (1) Let {C + t : t ∈ T } be a packing of translates of a triangular disc C. Then for each translate C + t there is an associated set At such that: (i) diam At ≤ 3diamC. (ii) The sets At ∪ (C + t), t ∈ T, do not overlap. (iii) The density of C + t in At ∪ (C + t), i.e. the quotient of the areas of these sets, equals the density of a certain lattice packing of C. Given C + t, consider translates C + r, C + s, not necessarily belonging to the given packing, such that the bottom vertices of C + s and C + r coincide with the left and right vertices of C + t. The translates C +r, C + s, C + t enclose a region A. We distinguish two cases (Fig. 30.2): First, no translate C + u, u ∈ T, overlaps A. Then put At = A. The region At is contained in a triangle, a congruent copy of which is contained in C + t. Hence Further, (2) diam At ≤ diam C. (3) each point of At is connected to a point of C + t by a vertical line segment and this line segment does not meet any translate C + u, u ∈ T \{t}. a b f C c Fig. 30.1. Triangular disc e d
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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
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24 The Minkowski-Hlawka Theorem 389
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The Variance Theorem of Rogers and
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The Selection Theorem of Mahler [68
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26 The Torus Group E d /L 395 where
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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 and 450: 30 Packing of Convex Bodies 439 The
Page 451 and 452: 30 Packing of Convex Bodies 441 Nex
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: A Lower Estimate for the Maximum La
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
Page 501 and 502: Generalizations 33 Optimum Quantiza
Page 503 and 504: 33 Optimum Quantization 493 set wit
Page 505 and 506: 33 Optimum Quantization 495 In (4)
Page 507 and 508: 33 Optimum Quantization 497 encoder
Page 509 and 510: 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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34 Koebe’s Representation Theorem
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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.,
Page 559 and 560:
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
Page 565 and 566:
References 555 1042. Zassenhaus, H.
Page 567 and 568:
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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