Gruber P. Convex and Discrete Geometry

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404 Geometry of Numbers Proof. Put ϱ = ϱ(C, L) and µ = µ(C, L), and let F be a fundamental parallelotope of L.BytC for t ≥ 0wemeanthesettC + L. It follows from Proposition 26.4 that (3) m(tC) = V (F ∩ tC) for t ≥ 0. Using (3), we prove the following two assertions. First, (4) m(tC) = t d V (C) for 0 ≤ t ≤ ϱ. For 0 ≤ t ≤ ϱ the set lattice {tC + l : l ∈ L} is a packing. Hence its density V (tC)/d(L) = t d V (C)/d(L) is the proportion of E d covered by the sets tC +l,see Sect. 30.1. The latter equals the proportion of F covered by the sets tC + l and thus is equal to V � F ∩ (tC + L) � /d(L) = V (F ∩ tC)/d(L). Hence (3) yields (4). Second, (5) µ = inf � t > 0 : m(tC) = d(L) � . If t > µ, then {tC + l : l ∈ L} is a covering. Then tC = tC + L = E d and thus m(tC) = V (F) = d(L) by (3). If t < µ, then {tC + l : l ∈ L} is not a covering. In particular, the family tC + l : l ∈ L of translates of the (compact) convex body by vectors of the lattice L do not cover F. Hence m(tC) = V (F ∩tC) = V � F ∩ (tC + L) � < V (F) = d(L) by (3). The proof of (5) is now complete. Noting that (s + t)C ⊇ sC + tC, the sum theorem shows that (6) m � (s + t)C � ≥ min � m(sC) + m(tC), d(L) � for s, t ≥ 0. Having shown (4)–(6), the proof of the theorem is easy: By (4), (7) m(ϱC) = ϱ d V (C), m(r 1 d ϱC) = rϱ d V (C). Applying (6) several times, (7) and (1), it follows that m � ϱ(q + r 1 d )C � ≥ min � qm(ϱC) + m(r 1 d ϱC), d(L) � = min � ϱ d (q + r)V (C), d(L) � = d(L). This, together with (5), finally yields (2). ⊓⊔ 27 Special Problems in the Geometry of Numbers In this chapter we have so far outlined systematic features of the geometry of numbers. What about problems? In contrast to other areas of mathematics, in the geometry of numbers there is only a rather small number of basic particular problems. Roughly speaking, these are of two types. First, special problems of a more arithmetic character, including problems on forms of various types. A selection of these will be considered here. Second, sets of problems involving reduction, packing, covering and tiling. While reduction has arithmetic and geometric aspects, the other three sets of problems of the second group definitely are geometric. Reduction, packing, covering and tiling will be studied in the subsequent sections. In this section we consider the following special problems:
27 Special Problems in the Geometry of Numbers 405 The conjecture on the product of inhomogeneous linear forms DOTU-matrices Mordell’s inverse problem for the linear form theorem of Minkowski Minima of the Epstein zeta function Lattice points in large convex bodies A further group of special problems deals with homogeneous and inhomogeneous minima of indefinite quadratic forms. These problems have been solved, mainly through the efforts of the Indian school of the geometry of numbers of Bambah, Dumir and Hans-Gill and their students, see the survey by Bambah, Dumir and Hans- Gill [58]. While, in recent decades, much progress has been achieved for indefinite quadratic forms, there is not much advance visible on the other problems. Exceptions are the results of Narzullaev and Ramharter [765] on DOTU-matrices and the proof of McMullen [704] of the conjecture on the product of inhomogeneous linear forms for d = 6. For more detailed information, see the book of the author and Lekkerkerker [447] and the reports of Malyshev [682], Bambah [56], Bambah, Dumir and Hans-Gill [58] and Bayer-Fluckiger and Nebe [84] 27.1 The Product of Inhomogeneous Linear Forms and DOTU Matrices A special problem, which has attracted interest since Minkowski first studied the 2-dimensional case, is the conjecture on the product of inhomogeneous linear forms. In spite of numerous contributions, the general case is still open and there are doubts whether the conjecture is true generally. The conjecture is one of those seminal problems which, over a century, has generated numerous notions, problems and results of different types in number theory and, in particular, in the geometry of numbers. One such notion is that of DOTU-matrices. Tools used in this context and, in particular, tools to attack the conjecture range from algebraic topology to measure and algebraic number theory. In this section we state the conjecture, describe the main lines of attack, additional results and problems and make some remarks on DOTU-matrices. The reader who is interested in more precise information may wish to consult [447], Malyshev [682], Bambah [56], Bambah, Dumir and Hans-Gill [58] and Narzullaev and Ramharter [765] and the references in these sources. The Conjecture on the Product of Inhomogeneous Linear Forms The following conjecture has been attributed to Minkowski, but according to Dyson [281] it is not contained in his written work. Conjecture 27.1. Let l1,...,ld be d real linear forms in d variables such that the absolute value δ of their determinant is positive. Then, for any α1,...,αd ∈ R, there is a point u ∈ Z d such that: � � � � � �� l1(u) − α1 ··· ld(u) − αd � ≤ δ . 2d
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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
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 and 372: By definition of bi+1, Thus (6) yie
Page 373 and 374: 21 Lattices 363 A similar, slightly
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: The formula to calculate the Jordan
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
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
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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.,
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.
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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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