Gruber P. Convex and Discrete Geometry

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7 Valuations 121 For ε > 0 the body (1 − ε)C is a compact subset of int C. Hence there is a polybox A with (1 − ε)C ⊆ A ⊆ int C ⊆ C and therefore, A ⊆ C ⊆ 1 1 − ε A. Since elementary volume on L(B) is non-decreasing and homogeneous of degree d, we have, � 1 V (A) ≤ V 1 − ε A � 1 = V (A) (1 − ε) d = V (A) + O(ε)V (A) ≤ V (A) + O(ε)V (B) as ε → 0. This implies that C is Jordan measurable. Assume, second, that C is contained in a hyperplane. Since C is compact, it can be covered with polyboxes of arbitrarily small elementary volume. Thus C is Jordan measurable with V (C) = 0. ⊓⊔ Theorem 7.5. The volume on C is a simple, homogeneous of degree d, nondecreasing, continuous, rigid motion invariant valuation. For the proof we need an auxiliary result about rotations: Let SO(d) denote the special orthogonal group of E d , i.e. the group of the orthogonal transformations of E d with determinant 1. Let B ={b1,...,bd} be the standard basis of E d and denote by SO(d, B) the set of all transformations in SO(d) whichfixatleastd − 2 vectors of the basis B. The required result is then as follows; for d = 3 it reduces to a version of a well-known result of Euler. Lemma 7.1. Let A ∈ SO(d). Then there are transformations A1,...,An ∈ SO(d, B) such that A = A1 ···An. Proof. The result holds trivially in case d = 2 since then SO(2) = SO(2, B). Suppose now that d > 2 and that the result holds for dimension d − 1. Let A ∈ SO(d) but A �∈ SO(d, B). Letu = Abd and assume without loss of generality that u �= ±bd. Consider the orthogonal projection of u into Ed−1 (= lin{b1,...,bd−1}) and let v be its unit normalization. There is S ∈ SO(d) with Sv = bd−1 and Sbd = bd. Since u ∈ lin{v,bd}, wehaveSu ∈ lin{bd−1, bd}. Let T ∈ SO(d, B) be such that Tb1 = b1,...,Tbd−2 = bd−2 and TSu = bd. Then TSAbd = TSu = bd. Clearly, S, TSA∈ SO(d) and both fix bd. By the induction hypothesis, there are S1,...,Sl, T1,...,Tm ∈ SO(d, B) such that S = S1 ···Sl and TSA = T1 ···Tm. Then A = S −1 T −1 T1 ···Tm = S −1 l ···S −1 1 T −1 T1 ···Tm. This concludes the induction and the proof is complete. ⊓⊔ Proof of the Theorem. We first show that (4) V is a valuation on C.
122 Convex Bodies Let C, D ∈ C such that C ∪ D ∈ C. In order to show that (5) V (C) + V (D) ≤ V (C ∪ D) + V (C ∩ D), choose sequences (Pn), (Qn) in L(B) with Then Pn ⊆ C, V (Pn) → V (C) and Qn ⊆ D, V (Qn) → V (D) as n →∞. Pn ∪ Qn ⊆ C ∪ D, Pn ∩ Qn ⊆ C ∩ D. The fact that V is a valuation on L(B) and the definition of Jordan measure on C yield the following. V (Pn) + V (Qn) = V (Pn ∪ Qn) + V (Pn ∩ Qn) ≤ V (C ∪ D) + V (C ∩ D). Now, let n →∞to get (5). The reverse inequality (6) V (C) + V (D) ≥ V (C ∪ D) + V (C ∩ D) is obtained in a similar way by considering sequences (Rn), (Sn) in L(B) such that Rn ⊇ C, V (Rn) → V (C) and Sn ⊇ D, V (Sn) → V (D). Having shown (5) and (6), the proof of (4) is complete. The statement that (7) V is simple has been shown in the last part of the proof of Theorem 7.4 above, while the statement that (8) V is homogeneous of degree d and non-decreasing, follows from the corresponding properties of V on L(B) and the definition of Jordan measure on C. For the proof that (9) V is continuous, we have to show the following: (10) Let C, C1, C2, ···∈C such that C1, C2, ···→C. Then V (C1), V (C2), ···→ V (C). Assume first, that V (C) >0. Since volume and Hausdorff metric are translation invariant, we may assume that o ∈ int C. Choose ϱ>0 such that ϱB d ⊆ C. Since C1, C2, ···→C, we have the following: Let ε>0. Then the inclusions Cn ⊆ C + εB d , C ⊆ Cn + εB 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
Page 81 and 82: 70 Convex Bodies for all y ∈ C an
Page 83 and 84: 72 Convex Bodies the existence of s
Page 85 and 86: 74 Convex Bodies where r(·) is a s
Page 87 and 88: 76 Convex Bodies Hence x ∈ conv e
Page 89 and 90: 78 Convex Bodies N = � � X, O Y
Page 91 and 92: 80 Convex Bodies Further topics of
Page 93 and 94: 82 Convex Bodies We are now ready t
Page 95 and 96: 84 Convex Bodies similarly, D j is
Page 97 and 98: 86 Convex Bodies For the proof of t
Page 99 and 100: 88 Convex Bodies The definition of
Page 101 and 102: 90 Convex Bodies If ei ∈ Pi is no
Page 103 and 104: 92 Convex Bodies Let v(·) denote (
Page 105 and 106: 94 Convex Bodies The present sectio
Page 107 and 108: 96 Convex Bodies Proposition 6.5. L
Page 109 and 110: 98 Convex Bodies Thus Second, Minko
Page 111 and 112: 100 Convex Bodies The Valuation Pro
Page 113 and 114: 102 Convex Bodies Minkowski’s the
Page 115 and 116: 104 Convex Bodies To remedy the sit
Page 117 and 118: 106 Convex Bodies Proof. Applying S
Page 119 and 120: 108 Convex Bodies Third, let C be a
Page 121 and 122: 110 Convex Bodies A Problem of Sant
Page 123 and 124: 112 Convex Bodies sets of S. Ifφ c
Page 125 and 126: 114 Convex Bodies = � (−1) |J|
Page 127 and 128: 116 Convex Bodies If P has dimensio
Page 129 and 130: 118 Convex Bodies Since, by Volland
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Page 135 and 136: 124 Convex Bodies P Q . . . . Fig.
Page 137 and 138: 126 Convex Bodies 7.3 Characterizat
Page 139 and 140: 128 Convex Bodies and the Bi have p
Page 141 and 142: 130 Convex Bodies Since the Riemann
Page 143 and 144: 132 Convex Bodies Seventh, o v3 v2
Page 145 and 146: 134 Convex Bodies Mean Width and th
Page 147 and 148: 136 Convex Bodies � χ(C ∩ mD)
Page 149 and 150: 138 Convex Bodies In the following,
Page 151 and 152: 140 Convex Bodies 7.5 Hadwiger’s
Page 153 and 154: 142 Convex Bodies 8.1 The Classical
Page 155 and 156: 144 Convex Bodies C ∩ H(tC ) C D
Page 157 and 158: 146 Convex Bodies Theorem 8.4. Let
Page 159 and 160: 148 Convex Bodies In the following
Page 161 and 162: 150 Convex Bodies Integration impli
Page 163 and 164: 152 Convex Bodies As a consequence
Page 165 and 166: 154 Convex Bodies Proof. � 1 2 w(
Page 167 and 168: 156 Convex Bodies Let ϱ>0 be the m
Page 169 and 170: 158 Convex Bodies Wulff’s Theorem
Page 171 and 172: 160 Convex Bodies Third, by (8) and
Page 173 and 174: 162 Convex Bodies Since the measure
Page 175 and 176: 164 Convex Bodies An application of
Page 177 and 178: 166 Convex Bodies Since f has Lipsc
Page 179 and 180: 168 Convex Bodies Clearly, the inte
Page 181 and 182: 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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186 Convex Bodies Theorem 9.10. Let
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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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210 Convex Bodies Theorem 11.4. Let
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212 Convex Bodies the maximum of th
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214 Convex Bodies (20) mni ≥ 1 λ
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216 Convex Bodies Prove analogous r
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218 Convex Bodies Heuristic Princip
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220 Convex Bodies λC + x νC + z
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222 Convex Bodies To show that (K +
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224 Convex Bodies The integral here
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226 Convex Bodies In the following
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228 Convex Bodies The first such re
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230 Convex Bodies v o Fig. 12.3. Sy
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232 Convex Bodies What Does the Bou
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234 Convex Bodies Corollary 13.1. L
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236 Convex Bodies Hence ν = 0 and
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238 Convex Bodies geometry, see Sei
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240 Convex Bodies We distinguish th
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242 Convex Bodies by Proposition 13
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244 Convex Polytopes The material i
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246 Convex Polytopes V-Polytopes an
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248 Convex Polytopes Convex cones w
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250 Convex Polytopes Generalized Co
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252 Convex Polytopes d 2 inequaliti
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254 Convex Polytopes The Face Latti
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256 Convex Polytopes Then (6) impli
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258 Convex Polytopes P in F. For an
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260 Convex Polytopes Euler’s Poly
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262 Convex Polytopes bijection betw
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264 Convex Polytopes The Converse o
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266 Convex Polytopes (geometric) fa
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268 Convex Polytopes consists preci
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270 Convex Polytopes The Lower and
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272 Convex Polytopes A problem of S
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274 Convex Polytopes The first cond
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276 Convex Polytopes The existence
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278 Convex Polytopes For d = 3 it f
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280 Convex Polytopes 16 Volume of P
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282 Convex Polytopes by (2). Hence
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284 Convex Polytopes (12) The volum
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286 Convex Polytopes T ∩ Z, respe
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288 Convex Polytopes An Alternative
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290 Convex Polytopes P clearly can
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292 Convex Polytopes 17 Rigidity Ri
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294 Convex Polytopes Fig. 17.2. Cau
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296 Convex Polytopes Call the index
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298 Convex Polytopes of V. Fori ∈
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300 Convex Polytopes Thus (3) impli
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302 Convex Polytopes Proof. (i)⇒(
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304 Convex Polytopes For other pert
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306 Convex Polytopes where λ is a
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308 Convex Polytopes 18.3 The Isope
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310 Convex Polytopes 19 Lattice Pol
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312 Convex Polytopes Embed E d into
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314 Convex Polytopes To see this, n
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316 Convex Polytopes The Coefficien
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318 Convex Polytopes If the triangl
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320 Convex Polytopes � qP(t) = No
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322 Convex Polytopes Now, take into
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324 Convex Polytopes L � P + (n +
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326 Convex Polytopes V (S) = V .Let
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328 Convex Polytopes (7) ψ|H d = 0
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330 Convex Polytopes Theorem 19.6.
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332 Convex Polytopes (15) L(nP) = L
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334 Convex Polytopes have to be sin
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336 Convex Polytopes economics, von
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338 Convex Polytopes Existence of S
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340 Convex Polytopes Description of
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342 Convex Polytopes (6) u piai xq
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344 Convex Polytopes For x ∈ Bd
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346 Convex Polytopes Rational Polyh
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348 Convex Polytopes Totally Dual I
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350 Convex Polytopes of the vectors
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Geometry of Numbers and Aspects of
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Extension to Crystallographic Group
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Relations Between Different Bases 2
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21 Lattices 359 triangle conv{o,(u,
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By definition of bi+1, Thus (6) yie
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21 Lattices 363 A similar, slightly
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21.4 Polar Lattices 21 Lattices 365
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The First Fundamental Theorem 22 Mi
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22 Minkowski’s First Fundamental
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If 22 Minkowski’s First Fundament
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22 Minkowski’s First Fundamental
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Proof. Consider the linear forms l1
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Proof. It is sufficient to consider
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Thus V (C) ≥ V (O) = 2d � �
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Given C and L, the problem arises t
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23 Successive Minima 383 as require
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24 The Minkowski-Hlawka Theorem 24
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(6) (V (J) ≈) � n�= 0 (7) J
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24 The Minkowski-Hlawka Theorem 389
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The Variance Theorem of Rogers and
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The Selection Theorem of Mahler [68
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26 The Torus Group E d /L 395 where
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Proposition 26.2. E d /L is a compa
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26 The Torus Group E d /L 399 theor
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26 The Torus Group E d /L 401 The m
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The formula to calculate the Jordan
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27 Special Problems in the Geometry
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Asymptotic Estimates 27 Special Pro
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27 Special Problems in the Geometry
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28 Basis Reduction and Polynomial A
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(ii) �b1� ≤2 1 2 (d−1) min
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28 Basis Reduction and Polynomial A
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and so 28 Basis Reduction and Polyn
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Shortest Lattice Vector Problem 28
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and therefore �x − m� 2 = �
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29 Packing of Balls and Positive Qu
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29.3 Error Correcting Codes and Bal
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30 Packing of Convex Bodies 439 The
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30 Packing of Convex Bodies 441 Nex
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Packing and Central Symmetrization
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Existence of Densest Lattice Packin
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30 Packing of Convex Bodies 447 Rem
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A Lower Estimate for the Maximum La
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Lattice Packing and Packing of Tran
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a f b o e 2D c Fig. 30.3. Disc and
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31 Covering with Convex Bodies 455
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32 Tiling with Convex Polytopes 463
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Define 32 Tiling with Convex Polyto
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33.1 Fejes Tóth’s Inequality for
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Noting that 2h 2 v − hhvv = 2π 2
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33 Optimum Quantization 485 The con
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(7) � (x, y) : x ∈ K, 0 ≤ y
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Next, we prove that (18) lim inf n
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Generalizations 33 Optimum Quantiza
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33 Optimum Quantization 493 set wit
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33 Optimum Quantization 495 In (4)
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33 Optimum Quantization 497 encoder
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34 Koebe’s Representation Theorem
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G 34 Koebe’s Representation Theor
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Di+1 34 Koebe’s Representation Th
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References 1. Achatz, H., Kleinschm
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References 515 36. Arias de Reyna,
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References 517 88. Beer, G., Topolo
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References 519 136. Bohr, H., Molle
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References 521 190. Carathéodory,
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References 523 237. Dantzig, G.B.,
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References 525 286. Edmonds, A.L.,
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References 527 337. Florian, A., Ex
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References 529 385. Goodman, J.E.,
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References 531 435. Gruber, P.M., C
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References 533 487. Heil, E., Über
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References 535 534. Hyuga, T., An e
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References 537 582. Khovanskiĭ (Ho
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References 539 632. Lazutkin, V.F.,
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References 541 684. Mani-Levitska,
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References 543 736. Minkowski, H.,
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References 545 783. Pach, J., Agarw
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References 547 835. Rickert, N.W.,
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References 549 891. Schmidt, E., Di
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References 551 941. Skubenko, B.F.,
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References 553 992. Teissier, B., V
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References 555 1042. Zassenhaus, H.
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558 INDEX Bravais classification of
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560 INDEX existence of elementary v
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562 INDEX Kneser-Macbeath theorem,
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564 INDEX rational lattice, 414 rat
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566 INDEX transference theorem of K
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568 AUTHOR INDEX Björner, 265 Blas
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570 AUTHOR INDEX Giles, J., 1 Giles
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572 AUTHOR INDEX Kuczma, 13, 17 Kü
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574 AUTHOR INDEX Pick, 316 Pisier,
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576 AUTHOR INDEX Tanemura, 464 Tarj
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578 List of Symbols sch, schL Schwa
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289. Manin: Gauge Field Theory and
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