Gruber P. Convex and Discrete Geometry

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460 Geometry of Numbers After these preparations the following will be shown first: (4) There is a set X ={x1,...,xm} ⊆F such that the proportion of space left uncovered by the set C + L + X, that is by the family � C + l + x : l ∈ L, x ∈ X � ,isatmost � 1 − 1 sd �m . Let 1C be the characteristic function of C. It follows from (3) that the characteristic function of the set C +L, that is the union of the family {C +l : l ∈ L},is � � 1C(x − l) : l ∈ L � . Thus the set C + L + xi where xi ∈ F has characteristic function � � 1C(x − l − xi) : l ∈ L � . This shows that, for X ={x1,...,xm} ⊆F, the characteristic function of the set S = E d \(C + l + X) left uncovered by the set C + L + X, that is by the family � C + l + xi : l ∈ L, xi ∈ X � , is given by: 1S(x) = m� � 1 − � � 1C(x − l − xi) . i=1 l∈L 1S is periodic with respect to L. Thus the proportion of space left uncovered by the set C + L + X equals V (S ∩ F) V (F) � 1 = V (F) F 1S(x) dx. The mean value of this proportion extended over all choices X ={x1,...,xm} of m points in F is thus 1 V (F) m � F � ··· � � 1 V (F) � 1S(x) dx dx1 ···dxm F 1 = V (F) F m+1 � F � � � m� � ··· 1 − F F i=1 � � = � � 1C(x − l − xi) dx1 ···dxm dx l∈L � �m � � 1 − � � � 1C(x − l − xi) dxi dx = = = = 1 V (F) m+1 1 V (F) m+1 1 V (F) m+1 1 V (F) m+1 F � F � F � F i=1 i=1 F l∈L m� � V (F) − � � � 1C(x − l − xi)dxi dx m� i=1 m� i=1 l∈L F � V (F) − � � V (F) − � l∈L F−x+l � E d 1C(−y)dy 1 V (F) m+1 V (F)� V (F) − V (C) � m = � 1C(−y)dy dx � 1 − � dx �m � V (C) = 1 − V (F) 1 sd �m
31 Covering with Convex Bodies 461 by (1). Since the mean value extended over all choices of m points in F of the proportion in question is (1 − 1/s d ) m , there is at least one choice X ={x1,...,xm} of m points in F for which this proportion is at most (1 − 1/s d ) m , concluding the proof of (4). Second, we prove the following proposition where the set X is as in (4). (5) Let 0 < t ≤ 1/d and Y ={y1,...,yn} ⊆F with n maximum such that the family � − tC + l + y : l ∈ L, y ∈ Y � is a packing contained in E d \(C + L + X). Then n ≤ sd td � 1 − 1 sd �m . This is easy. First, the proportion of space covered by the packing � −tC+l + y : l ∈ L, y ∈ Y � , that is its density, is nt d V (C)/V (F) = nt d /s d by (1) and the choice of L = s Z d . Second, this packing is contained in E d \(C + L + X) and the proportion of space covered by the latter set is at most (1 − 1/s d ) m by (4). The proof of (5) is complete. Third, note (4) and (5). In order to prove that (6) � (1 + t)C + l + z : l ∈ L, z ∈ X ∪ Y � is a covering, let w ∈ Ed . We have to show that w is contained in at least one of the bodies in (6). By (2), (3) and since by (5) 0 < t ≤ 1 d , the bodies −tC+m+w, m ∈ L, are pairwise disjoint. We distinguish two cases. First, (−tC + L + w) ∩ (C + L + X) �= ∅. Then there are p, q ∈ C, l, m ∈ L, xi ∈ X such that −tp + l + w = q + m + xi, or w = tp + q − l + m + xi ∈ (1 + t)C + L + X. Second, if the first alternative does not hold, then −tC + L + w ⊆ Ed � � \(C + L + X). By the maximality of the family − tC + l + y : l ∈ L, y ∈ Y we then have (−tC + L + w) ∩ (−tC + L + Y ) �= ∅. Thus there are p, q ∈ C, l, m ∈ L, y j ∈ Y such that −tp+l + w =−td+ m + y j , or w = tp + tq − l + m + y j ∈ (1 + t)C + L + Y by (2) and since 0 < t ≤ 1/d by (5). This concludes the proof of (6). Fourth, noting (1), the definition of L = s Zd , Propositions (4) and (5) show that (7) the density of the covering in (6) is (1 + t) d (m + n) s d ≤ (1 + t) d � m 1 + sd td � 1 − 1 sd �m� . By choosing s, m, t suitably this finally yields the desired upper bound for a covering of E d by translates of (1 + t)C, and thus for a corresponding covering of E d by translates of C:Let m 1 = d log sd t , t = 1 d log 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
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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
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 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 467 and 468: 31 Covering with Convex Bodies 457
Page 469: 31 Covering with Convex Bodies 459
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
Page 511 and 512: G 34 Koebe’s Representation Theor
Page 513 and 514: 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.,
Page 537 and 538:
References 527 337. Florian, A., Ex
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References 529 385. Goodman, J.E.,
Page 541 and 542:
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
Page 587 and 588:
578 List of Symbols sch, schL Schwa
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289. Manin: Gauge Field Theory and
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