Gruber P. Convex and Discrete Geometry

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466 Geometry of Numbers F G p q H Fig. 32.2. Dirichlet–Voronoĭ tilings are facet-to-facet For the proof of (2) it is sufficient to show the following: if K is a cube, then only finitely many of the Dirichlet–Voronoĭ cells D(p) meet K and these form a tiling of K .Letq ∈ D. Since D is discrete, all but finitely many of the halfspaces {x : �x − p� ≤�x −q�} are disjoint from K . Thus, only finitely many Dirichlet–Voronoĭ cells D(p) meet K ,sayD(p1),...,D(pk). Any two of these cells are separated by a hyperplane and thus have disjoint interiors. Since D is discrete, for each point of E d there is a least one nearest point in D. Thus E d is the union of all cells. Together, this implies that � D(p1) ∩ K,...,D(pk) ∩ K � form a tiling of K . The third and final step is to show the following: (3) The Dirichlet–Voronoĭ cells {D(p) : p ∈ D} form a facet-to-facet tiling of E d . By (1) and (2), each facet of a cell D(p), p ∈ D, is covered by facets of other such cells. Thus, if (3) did not hold, there are points p, q ∈ D such that the tiles D(p) and D(q) have facets F and G, respectively, which overlap, but do not coincide. Then there is a (d − 2)-dimensional face H of G, say, which meets the relative interior of F. Project F, G, H onto a 2-dimensional plane orthogonal to H. Then we have the configuration of Fig. 32.2. Since H is the intersection of two facets of D(q), one of which is G, there is a point r ∈ D which, together with q, determines this facet. Then the hyperplane {x :�x − p� =�x − r�} cuts off the part of F outside H. Thus this part is not contained in D(p). This contradiction concludes the proof of (3). Having shown (1)–(3), the proof of the proposition is complete. ⊓⊔ Delone Sets, the Empty Sphere Method and Delone Triangulations We follow Delone [255], see also Delone and Ryshkov [259]. Let D be an (r, R)system or Delone set in E d where r, R > 0. That is, a discrete set D in E d such that any two distinct points have distance at least r and for any point in E d there is a point in D at distance at most R. Lattices and periodic sets as considered by Zassenhaus, are Delone sets but not vice versa. Delone sets still have a certain uniformity and regularity. Now the aim is to construct a tiling of E d with convex polytopes such that the vertices of these polytopes are precisely the points of D. Particular such tilings r
32 Tiling with Convex Polytopes 467 are called Delone triangulations or Delone tilings. These can be defined by Delone’s empty sphere method: Consider all Euclidean spheres such that D contains no point in the interior but d + 1 or more of its points are on the sphere and such that these points are not contained in a hyperplane. For each such empty sphere take the convex hull of the points of D on it. This gives a proper convex polytope. Proposition 32.2. Let D be a Delone set. Then the proper convex polytopes obtained by the empty sphere method form a facet-to-facet tiling of E d . Proof. We first show the following: (4) Let P, Q be two distinct convex polytopes obtained by the empty sphere method. Then int P ∩ int Q =∅. Let S, T be the corresponding empty spheres. None of these can contain the other one. If they are disjoint or touch, we are done. If not, they intersect along a common (d −2)-sphere. The hyperplane H through the latter cuts off from each of the spheres S, T a spherical cap contained in the interior of the other sphere. Thus none of these caps can contain a point of D. This shows that H separates the points of D ∩ S from the points of D ∩ T , which, in turn, implies (4). Next, we prove the following: (5) Let P be a polytope obtained by the empty sphere method from D and F a facet of P. Then there is another such polytope which meets P in F. Let S be the empty sphere corresponding to P. Move the centre of S in the direction of the exterior normal of F while keeping the vertices of F on the sphere. By the definition of D, this will eventually lead to an empty sphere T which contains the vertices of F and one or more points of D on the far side of F. The polytope corresponding to T is the desired polytope which meets P in F. The last step is to show that the following holds: (6) The polytopes obtained by the empty sphere method from D cover E d . Since the centre of an empty sphere has distance at most R from the nearest point of D, its radius is at most R. Hence each of the polytopes obtained by the empty sphere method has diameter at most 2R. The vertices of each of these polytopes are contained in D. Since D is discrete, each bounded set meets only finitely many such polytopes. If (6) did not hold, connect a point of the complement with a line segment which avoids all faces of our polytopes of dimension at most d − 2 to a point in the interior of one of the polytopes. The first point where this line segment meets a polytope, say P, is an interior point of a facet F of P and there is no polytope which meets P in F. Since this contradicts (5), the proof of (6) is complete. The proposition now follows from (4) to (6). ⊓⊔ An Alternative way to Construct Dirichlet–Voronoi and Delone Tilings Let D be a general discrete set or a Delone set. Embed E d into E d+1 as usual (first d coordinates). Consider, in E d+1 , the solid paraboloid
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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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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 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 475: 32 Tiling with Convex Polytopes 465
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
Page 511 and 512: G 34 Koebe’s Representation Theor
Page 513 and 514: Di+1 34 Koebe’s Representation Th
Page 523 and 524: References 1. Achatz, H., Kleinschm
Page 525 and 526: 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
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
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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
Page 587 and 588:
578 List of Symbols sch, schL Schwa
Page 589:
289. Manin: Gauge Field Theory and
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