Gruber P. Convex and Discrete Geometry

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XII Contents 20 Linear Optimization . ....................................... 335 20.1 Preliminaries and Duality . ............................ 336 20.2 TheSimplexAlgorithm............................... 339 20.3 The Ellipsoid Algorithm . . ............................ 343 20.4 Lattice Polyhedra and Totally Dual Integral Systems ....... 345 20.5 HilbertBasesandTotallyDualIntegralSystems .......... 348 Geometry of Numbers and Aspects of Discrete Geometry ............... 353 21 Lattices................................................... 355 21.1 Basic Concepts and Properties and a Linear Diophantine Equation ........................................... 356 21.2 CharacterizationofLattices............................ 359 21.3 Sub-Lattices . ....................................... 361 21.4 PolarLattices ....................................... 365 22 Minkowski’s First Fundamental Theorem . ..................... 366 22.1 The First Fundamental Theorem . . . ..................... 366 22.2 Diophantine Approximation and Discriminants of Polynomials . . . . . . ................................ 370 23 Successive Minima . . ....................................... 375 23.1 Successive Minima and Minkowski’s Second Fundamental Theorem ........................................... 376 23.2 Jarník’s Transference Theorem and a Theorem of Perron andKhintchine ...................................... 380 24 The Minkowski–Hlawka Theorem ............................ 385 24.1 The Minkowski–Hlawka Theorem . ..................... 385 24.2 Siegel’s Mean Value Theorem and the Variance Theorem of Rogers–Schmidt . . . ................................ 388 25 Mahler’s Selection Theorem . ................................ 391 25.1 Topology on the Space of Lattices . ..................... 391 25.2 Mahler’s Selection Theorem . . . . . . ..................... 392 26 The Torus Group E d /L ...................................... 395 26.1 Definitions and Simple Properties of E d /L ............... 395 26.2 The Sum Theorem of Macbeath–Kneser . . . .............. 398 26.3 Kneser’s Transference Theorem . . . ..................... 403 27 Special Problems in the Geometry of Numbers . . . . .............. 404 27.1 The Product of Inhomogeneous Linear Forms and DOTU Matrices............................................ 405 27.2 Mordell’s Inverse Problem and the Epstein Zeta-Function . . 408 27.3 Lattice Points in Large Convex Bodies . . . . .............. 410 28 Basis Reduction and Polynomial Algorithms . . . . . . .............. 411 28.1 LLL-Basis Reduction . ................................ 411 28.2 Diophantine Approximation, the Shortest and the Nearest Lattice Vector Problem . . . ............................ 417
Contents XIII 29 Packing of Balls and Positive Quadratic Forms . . . . .............. 421 29.1 Densest Lattice Packing of Balls in Dimensions 2 and 3 .... 422 29.2 Density Bounds of Blichfeldt and Minkowski–Hlawka . . . . . 424 29.3 Error Correcting Codes and Ball Packing . . .............. 427 29.4 Geometry of Positive Definite Quadratic Forms and Ball Packing . ........................................... 430 30 Packing of Convex Bodies . . . ................................ 439 30.1 Definitions and the Existence of Densest Lattice Packings . . 440 30.2 Neighbours . . ....................................... 445 30.3 The Betke–Henk Algorithm and the Lower Bound for δL(C) ofMinkowski–Hlawka....................... 448 30.4 Lattice Packing Versus Packing of Translates . . . .......... 450 31 CoveringwithConvexBodies ................................ 454 31.1 Definitions, Existence of Thinnest Lattice Coverings and the Covering Criterion of Wills ..................... 455 31.2 StarNumbers ....................................... 457 31.3 Rogers’s Upper Bound for ϑT (C) ...................... 458 31.4 Lattice Covering Versus Covering with Translates . . . . . . . . . 462 32 Tiling with Convex Polytopes ................................ 463 32.1 Dirichlet–Voronoĭ and Delone Tilings and Polyhedral Complexes.......................................... 463 32.2 The Venkov–Alexandrov–McMullen Characterization ofTranslativeTiles................................... 470 32.3 Conjectures and Problems of Voronoĭ, Hilbert, Minkowski, andKeller .......................................... 478 33 OptimumQuantization...................................... 480 33.1 Fejes Tóth’s Inequality for Sums of Moments . . .......... 481 33.2 Zador’s Asymptotic Formula for Minimum Distortion . . . . . 484 33.3 Structure of Minimizing Configurations, and a Heuristic Principle ........................................... 492 33.4 Packing and Covering of Circles, Data Transmission andNumericalIntegration............................. 494 34 Koebe’s Representation Theorem for Planar Graphs .............. 499 34.1 The Extension of Koebe’s Theorem by Brightwell and Scheinerman . . . . ................................ 500 34.2 Thurston’s Algorithm and the Riemann Mapping Theorem . 509 References ...................................................... 513 Index ........................................................... 567 Author Index .................................................... 577
Page 1 and 2: Grundlehren der mathematischen Wiss
Page 3 and 4: Peter M. Gruber Institute of Discre
Page 5 and 6: VI Preface theory of finite groups
Page 7 and 8: Contents Preface ..................
Page 9: Contents XI 12 Special Convex Bodie
Page 13 and 14: 2 Convex Functions 1 Convex Functio
Page 15 and 16: 4 Convex Functions x epi f f C y Fi
Page 17 and 18: 6 Convex Functions Together these i
Page 19 and 20: 8 Convex Functions Theorem 1.4. Let
Page 21 and 22: 10 Convex Functions z = f (x) + f
Page 23 and 24: 12 Convex Functions 1.3 Convexity C
Page 25 and 26: 14 Convex Functions Proof (by affin
Page 27 and 28: 16 Convex Functions Minkowski’s I
Page 29 and 30: 18 Convex Functions (ii) Let x > 0.
Page 31 and 32: 20 Convex Functions 2 Convex Functi
Page 33 and 34: 22 Convex Functions | f (z) − f (
Page 35 and 36: 24 Convex Functions First-Order Dif
Page 37 and 38: 26 Convex Functions Remark. More pr
Page 39 and 40: 28 Convex Functions even more is tr
Page 41 and 42: 30 Convex Functions G(x). The proof
Page 43 and 44: 32 Convex Functions by (15), unifor
Page 45 and 46: 34 Convex Functions 2.4 A Stone-Wei
Page 47 and 48: 36 Convex Functions x x Fig. 2.1. S
Page 49 and 50: 38 Convex Functions F is convex. Us
Page 51 and 52: 40 Convex Bodies results, character
Page 53 and 54: 42 Convex Bodies Convex Hulls Given
Page 55 and 56: 44 Convex Bodies The next result is
Page 57 and 58: 46 Convex Bodies C ∩ L ⊥ clearl
Page 59 and 60: 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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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.,
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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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