Gruber P. Convex and Discrete Geometry

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362 Geometry of Numbers (ii) Given a basis {b1,...,bd} of L, there is a basis {c1,...,cd} of M such that (1) holds. Proof. (i) This is an immediate consequence of the case i = d of Proposition (3) in the proof of Theorem 21.2. (ii) We first show that (2) uL ⊆ M with a suitable positive u ∈ Z. Consider bases {b1,...,bd} of L and {c1,...,cd} of M such that (1) holds. The inverse of the integer lower triangular matrix (uik), with determinant u = u11 ··· udd �= 0, is a lower triangular matrix, the entries of which are of the form vik/u, where vik ∈ Z. Hence ub1 = v11c1 ub2 = v21c1 + v22c2 ............................ ubd = vd1c1 +······+vddcd. From this it follows that uL is a sub-lattice of M, concluding the proof of (2). To prove (ii), let {b1,...,bd} be a basis of L. Choose u as in (2). Then {ub1,...,ubd} is a basis of uL. An application of (i) to the sub-lattice uL of M thus yields a basis {d1,...,dd} of M such that (3) ub1 = w11d1 ub2 = w21d1 + w22d2 ............................. ubd = wd1d1 +······+wdddd where wik ∈ Z,wii �= 0. Solving (3) for d1,...,dd, we see that we may express d1,...,dd in the form (1) with d1,...,dd instead of c1,...,cd, but where the uik are rational. Since {b1,...,bd} is a basis of L and M a sub-lattice, the uik are, in fact, integers. ⊓⊔ Primitive Points and Bases If {b1,...,bd} is a basis of L and b = u1b1 +···+udbd, then b is primitive if and only if 1 is the greatest common divisor of u1,...,ud. Corollary 21.1. Let b be a primitive point of a lattice L in E d . Then there are points b2,...,bd ∈ L, such that {b, b2,...,bd} is a basis of L. Proof. Choose c2,...,cd ∈ L, such that b, c2,...,cd are linearly independent. By Proposition (i) of the above theorem, there is a basis {b1,...,bd} of L, such that b = u11b1 c2 = u21b1 + u22b2 ........................... cd = ud1b1 +······+uddbd where uik ∈ Z, uii �= 0. Since b is primitive, this can hold only if u11 =±1orb =±b1. Hence {b, b2,...,bd} is also a basis of L. ⊓⊔
21 Lattices 363 A similar, slightly more complicated proof leads to the following result: Corollary 21.2. Let b1,...,bk be k linearly independent points of a lattice L such that � α1b1 +···+αkbk : 0 ≤ αi < 1 � ∩ L ={o}. Then, there are points bk+1,...,bd ∈ L such that {b1,...,bd} is a basis of L. Closely Related Bases of a Lattice and a Sub-Lattice The following result shows that, given a sub-lattice M of a lattice L, there are bases of M and L which are related in a particularly simple way. We could ascertain whether this result is a consequence of a more general result on Abelian groups. Theorem 21.4. Let M be a sub-lattice of a lattice L in E d . Then there are bases {c1,...,cd} of M and {b1,...,bd} of L, such that c1 = u1b1,...,cd = udbd, where ui ∈ Z \{0}. Proof (by induction on d). If d = 1, the theorem is easy to see. Assume now that d > 1 and that the theorem holds for d − 1. The proof for d is divided into two steps. In the first step we treat a special case: (4) Let M be a sub-lattice of a lattice L in E d which contains a primitive point b1 of L. Then the theorem holds. In the following, we consider lower dimensional lattices in E d , but this should not cause difficulties. Since b1 is a primitive point of L, there is a basis of L of the form {b1, b2,...,bd} by Corollary 21.1. Then L ={ub1 + l : u ∈ Z, l ∈ lin{b2,...,bd}∩L = L ′ }. L ′ is a lattice of dimension d − 1inE d . Let a point m ∈ M ⊆ L be given. Then, since b1 ∈ M, we have m − ub1 ∈ lin{b2,...,bd}∩M for suitable u ∈ Z. Hence M ={ub1 + n : u ∈ Z, n ∈ lin{b2,...,bd}∩M = M ′ }. M ′ is a (d − 1)-dimensional sub-lattice of the (d − 1)-dimensional lattice L ′ .By induction, there are bases {b2,...,bd} of L ′ and {c2,...,cd} of M ′ , such that c2 = u2b2,...,cd = udbd for suitable ui ∈ Z. The bases {b1, b2,...,bd} of L and {b1, c2,...,cd} of M are then of the desired form, concluding the proof for d in the special case (4). In the second step, we consider a general sub-lattice M of L. For each point m ∈ M \{o}, let j be the unique positive integer such that m = jl, where l ∈ L is primitive. Call j the index of m. Choose c ∈ M \{o}, such that c has minimum
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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
Page 321 and 322: 310 Convex Polytopes 19 Lattice Pol
Page 323 and 324: 312 Convex Polytopes Embed E d into
Page 325 and 326: 314 Convex Polytopes To see this, n
Page 327 and 328: 316 Convex Polytopes The Coefficien
Page 329 and 330: 318 Convex Polytopes If the triangl
Page 331 and 332: 320 Convex Polytopes � qP(t) = No
Page 333 and 334: 322 Convex Polytopes Now, take into
Page 335 and 336: 324 Convex Polytopes L � P + (n +
Page 337 and 338: 326 Convex Polytopes V (S) = V .Let
Page 339 and 340: 328 Convex Polytopes (7) ψ|H d = 0
Page 341 and 342: 330 Convex Polytopes Theorem 19.6.
Page 343 and 344: 332 Convex Polytopes (15) L(nP) = L
Page 345 and 346: 334 Convex Polytopes have to be sin
Page 347 and 348: 336 Convex Polytopes economics, von
Page 349 and 350: 338 Convex Polytopes Existence of S
Page 351 and 352: 340 Convex Polytopes Description of
Page 353 and 354: 342 Convex Polytopes (6) u piai xq
Page 355 and 356: 344 Convex Polytopes For x ∈ Bd
Page 357 and 358: 346 Convex Polytopes Rational Polyh
Page 359 and 360: 348 Convex Polytopes Totally Dual I
Page 361 and 362: 350 Convex Polytopes of the vectors
Page 363 and 364: Geometry of Numbers and Aspects of
Page 365 and 366: Extension to Crystallographic Group
Page 367 and 368: Relations Between Different Bases 2
Page 369 and 370: 21 Lattices 359 triangle conv{o,(u,
Page 371: By definition of bi+1, Thus (6) yie
Page 375 and 376: 21.4 Polar Lattices 21 Lattices 365
Page 377 and 378: The First Fundamental Theorem 22 Mi
Page 379 and 380: 22 Minkowski’s First Fundamental
Page 381 and 382: If 22 Minkowski’s First Fundament
Page 383 and 384: 22 Minkowski’s First Fundamental
Page 385 and 386: Proof. Consider the linear forms l1
Page 387 and 388: Proof. It is sufficient to consider
Page 389 and 390: Thus V (C) ≥ V (O) = 2d � �
Page 391 and 392: Given C and L, the problem arises t
Page 393 and 394: 23 Successive Minima 383 as require
Page 395 and 396: 24 The Minkowski-Hlawka Theorem 24
Page 397 and 398: (6) (V (J) ≈) � n�= 0 (7) J
Page 399 and 400: 24 The Minkowski-Hlawka Theorem 389
Page 401 and 402: The Variance Theorem of Rogers and
Page 403 and 404: The Selection Theorem of Mahler [68
Page 405 and 406: 26 The Torus Group E d /L 395 where
Page 407 and 408: Proposition 26.2. E d /L is a compa
Page 409 and 410: 26 The Torus Group E d /L 399 theor
Page 411 and 412: 26 The Torus Group E d /L 401 The m
Page 413 and 414: The formula to calculate the Jordan
Page 415 and 416: 27 Special Problems in the Geometry
Page 417 and 418: Asymptotic Estimates 27 Special Pro
Page 419 and 420: 27 Special Problems in the Geometry
Page 421 and 422: 28 Basis Reduction and Polynomial A
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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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