Gruber P. Convex and Discrete Geometry

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strong membership oracle, 60 strong separation oracle, 60 strongly separated sets, 58 sub-facet of a polytope, 471 successive minimum, 376 sum of moments, 481 sum theorem of Kneser–Macbeath, 399 support cone, 206 function, 56 halfspace, 54 hyperplane, 53, 55 set, 54 surface area formula of Cauchy, 106 symmetric difference metric, 209 symmetric with respect to a group, 168 symmetry condition of Minkowski, 307 theorem of Alexandrov, 198, 301 Alexandrov on differentiability, 28, 74 Alexandrov on projections, 125 Alexandrov, Fenchel-Jessen, 193 Anderson and Klee, 69 Bezout, 335 Birkhoff, 77 Blaschke, 84 Bohr–Mollerup, 18 Bruggesser-Mani, 266 Brunn–Minkowski, 142 Carathéodory, 43, 47 Choquet, 221 Courant–Hilbert, 36 Dehn, 290 Ehrhart, 311 Frobenius–König, 77 Gluck, 298 Groemer, 117 Gross, 172 Hadwiger, 132 Hahn–Banach, 22 Hartogs, 51 Helly, 47 INDEX 565 Jarník, 381 John, 204 Jung, 49 Kneser, 403 Kneser–Macbeath, 399 Lyapunov, 61 Mahler, 393 Mazur, 71 McMullen–Bernstein, 322 Minding–Kouchnirenko–Bernstein, 335 Minkowski, 304, 367 Minkowski on mixed volumes, 89 Minkowski on successive minima, 376 Minkowski–Hlawka, 385 Perles–Blind–Mani, 275 Perron–Khintchine, 383 Pogorelov, 199 Radon, 47 Reeve–Macdonald, 318 Reidemeister, 25 Richter-Gebert, 279 Rogers, 459 Rogers–Schmidt, 391 Schneider, 239 Siegel, 389 Steiner on parallel bodies, 93 Steinitz, 271 Swinnerton-Dyer, 447 Venkov–Alexandrov–McMullen, 474 Volland, 112, 115 Voronoĭ on extreme forms, 438 Wills, 457 Wulff, 158 Zador, 484 Thurston’s algorithm, 509 tiling of congruent copies, 464 time optimal control, 64 topological vector space, 221 topology on the space of lattices, 392 torsional rigidity, 180 torus group, 396 totally dual integral system, 348 transference theorem of Jarník, 381
566 INDEX transference theorem of Kneser, 403 translation invariant valuation, 119 translative covering density, 455 translative packing density, 441 translative tiling, 464 twice differentiable almost everywhere, 10 at a boundary point, 72 function, 28 typical elements of a Baire space, 70 underlying set of a polytopal complex, 266 uniformly distributed sequence of sets, 493 uniqueness of elementary volume of polytopes, 281 upper bound conjecture, 270 Urysohn’s inequality, 153 V-cone, 248 V-polyhedron, 247 V-polytope, 246 valuation, 110, 324 variance theorem of Rogers–Schmidt, 391 variation of a vector-valued measure, 61 vector-lattice, 220 vector-quantizer, 497 Venkov–Alexandrov–McMullen theorem, 474 vertex, 245 -country incidence graph, 501 of a framework, 298 of a graph, 261 of an abstract complex, 265 Volland’s first extension theorem, 112 Volland’s polytope extension theorem, 115 volume, 120 Voronoĭ’s conjecture on parallelohedra, 478 Voronoĭ’s theorem on extreme forms, 438 wave equation, 182 weak membership oracle, 60 weak separation oracle, 60 weights for numerical integration, 498 Weyl’s problem, 187, 198 Wulff’s theorem, 158 Young’s inequality, 15 Zador’s theorem, 484 Zassenhaus’s conjecture on covering, 463 Zassenhaus’s conjecture on packing, 454 zeta function, 408 zonoid, 129 zonotope, 129
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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
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26 The Torus Group E d /L 399 theor
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26 The Torus Group E d /L 401 The m
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The formula to calculate the Jordan
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27 Special Problems in the Geometry
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Asymptotic Estimates 27 Special Pro
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27 Special Problems in the Geometry
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28 Basis Reduction and Polynomial A
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(ii) �b1� ≤2 1 2 (d−1) min
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28 Basis Reduction and Polynomial A
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and so 28 Basis Reduction and Polyn
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Shortest Lattice Vector Problem 28
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and therefore �x − m� 2 = �
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29 Packing of Balls and Positive Qu
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29.3 Error Correcting Codes and Bal
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30 Packing of Convex Bodies 439 The
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30 Packing of Convex Bodies 441 Nex
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Packing and Central Symmetrization
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Existence of Densest Lattice Packin
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30 Packing of Convex Bodies 447 Rem
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A Lower Estimate for the Maximum La
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Lattice Packing and Packing of Tran
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a f b o e 2D c Fig. 30.3. Disc and
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31 Covering with Convex Bodies 455
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32 Tiling with Convex Polytopes 463
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Define 32 Tiling with Convex Polyto
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33.1 Fejes Tóth’s Inequality for
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Noting that 2h 2 v − hhvv = 2π 2
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33 Optimum Quantization 485 The con
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(7) � (x, y) : x ∈ K, 0 ≤ y
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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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Page 523 and 524: References 1. Achatz, H., Kleinschm
Page 525 and 526: References 515 36. Arias de Reyna,
Page 527 and 528: References 517 88. Beer, G., Topolo
Page 529 and 530: References 519 136. Bohr, H., Molle
Page 531 and 532: References 521 190. Carathéodory,
Page 533 and 534: References 523 237. Dantzig, G.B.,
Page 535 and 536: References 525 286. Edmonds, A.L.,
Page 537 and 538: References 527 337. Florian, A., Ex
Page 539 and 540: References 529 385. Goodman, J.E.,
Page 541 and 542: References 531 435. Gruber, P.M., C
Page 543 and 544: References 533 487. Heil, E., Über
Page 545 and 546: References 535 534. Hyuga, T., An e
Page 547 and 548: References 537 582. Khovanskiĭ (Ho
Page 549 and 550: References 539 632. Lazutkin, V.F.,
Page 551 and 552: References 541 684. Mani-Levitska,
Page 553 and 554: References 543 736. Minkowski, H.,
Page 555 and 556: References 545 783. Pach, J., Agarw
Page 557 and 558: References 547 835. Rickert, N.W.,
Page 559 and 560: References 549 891. Schmidt, E., Di
Page 561 and 562: 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
Page 569 and 570: 560 INDEX existence of elementary v
Page 571 and 572: 562 INDEX Kneser-Macbeath theorem,
Page 573: 564 INDEX rational lattice, 414 rat
Page 577 and 578: 568 AUTHOR INDEX Björner, 265 Blas
Page 579 and 580: 570 AUTHOR INDEX Giles, J., 1 Giles
Page 581 and 582: 572 AUTHOR INDEX Kuczma, 13, 17 Kü
Page 583 and 584: 574 AUTHOR INDEX Pick, 316 Pisier,
Page 585 and 586: 576 AUTHOR INDEX Tanemura, 464 Tarj
Page 587 and 588: 578 List of Symbols sch, schL Schwa
Page 589: 289. Manin: Gauge Field Theory and
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