Gruber P. Convex and Discrete Geometry

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9 Symmetrization 175 9.2 The Isodiametric, Isoperimetric, Brunn–Minkowski, Blaschke–Santaló and Mahler Inequalities If C is a proper convex body but not a ball, then a suitable Steiner symmetral of C has smaller isoperimetric quotient than C. This observation led Steiner [958, 959], erroneously, to believe that Steiner symmetrization shows that balls have minimum isoperimetric quotient. The gap in this proof of the isoperimetric inequality was filled by Blaschke. He showed that Steiner symmetrization together with the Blaschke selection theorem, or rather the sphericity theorem of Gross, also provide easy proofs of the isodiametric, the isoperimetric and the Brunn–Minkowski inequalities. Below we first present the proofs of Gross [407] and Blaschke [124] of these inequalities, but do not discuss the equality cases. Then the proof of Ball [47] and Meyer and Pajor [720] of the Blaschke–Santaló inequality will be given, again, excluding a discussion of the equality case. As a counterpart of the Blaschke–Santaló inequality, we present an inequality of Mahler. Let κd = V (B d ). The Isodiametric Inequality A simple proof using d Steiner symmetrizations yields the following result of Bieberbach [113], a refinement of which was given by Urysohn [1004]. For a different proof, see Sect. 8.3. Theorem 9.2. Let C ∈ C. Then V (C) ≤ 1 2 d (diam C)d V (B d ). Proof. By symmetrizing C with respect to each coordinate hyperplane we obtain a convex body D which is symmetric in the origin o and with V (D) = V (C) and diam D ≤ diam C, see Proposition 9.1(vi,viii). Then D ⊆ ( 1 2 diam D)Bd . ⊓⊔ The Isoperimetric Inequality An equally simple proof yields the next result. Theorem 9.3. Let C ∈ Cp. Then S(C) d V (C) d−1 ≥ S(Bd ) d V (B d . ) d−1 Proof. By Corollary 9.1, there are hyperplanes H1, H2,... through o, such that st Hn ···st H1C → � V (C) Now using Proposition 9.1(vii) and the continuity of the surface area S(·), see Theorem 6.13, we obtain the desired inequality: �� 1 V (C) d S(C) ≥ S(st H1C) ≥···→ S B d � = V (C) d−1 d S(B d ). ⊓⊔ κd κd � 1 d B d . κ d−1 d d
176 Convex Bodies The Brunn–Minkowski Inequality The following proof reduces the Brunn–Minkowski inequality to the trivial case where both bodies are balls. Theorem 9.4. Let C, D ∈ C. Then V (C + D) 1 d ≥ V (C) 1 d + V (D) 1 d . Proof. It is sufficient to consider the case where C, D ∈ Cp. By Corollary 9.1, there are hyperplanes H1, H2,... through o, such that st Hn ···st � V (C) � 1 d d H1C → B , st Hn ···st H1 D → κd � V (D) � 1 d d B . Thus Proposition 9.1(vi,iii) and the continuity of V , see Theorem 7.5, yield the following. V (C + D) 1 d = V � st H1 (C + D)� 1 d ≥ V (st H1C + st H1 D) 1 d ≥··· �� 1 V (C) d → V B d � � 1 V (D) d + B d � 1 ⎛ d = V ⎝ 1 � V (C) 1 d + V (D) 1� d κd = V (C) 1 d + V (D) 1 d . ⊓⊔ κd The Blaschke–Santaló Inequality Blaschke [126] proved the following result for d = 3. It was extended to general d by Santaló [880]. Theorem 9.5. Let C ∈ Cp be o-symmetric. Then κd V (C)V (C ∗ ) ≤ κ 2 d . Proof. By the Corollary 9.1 to the sphericity theorem of Gross, there are hyperplanes H1, H2,... through o, such that Cn = st Hn ···st � � 1 V (C) d H1C → B d . An easy argument for polar bodies then shows that C ∗ n → � � 1 κd d B V (C) d . Thus V (C)V (C ∗ ) ≤ V (st H1 C)V � (st H1 C)∗� = V (C1)V (C ∗ 1 ) κd κ 1 d d ≤ V (st H2C1)V � (st H2C1) ∗� = V (C2)V (C ∗ 2 ) ≤ ... → V (B d ) 2 = κ 2 d . by Proposition 9.2 and the continuity of volume on C, see Theorem 7.5. ⊓⊔ B d ⎞ ⎠ 1 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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Page 137 and 138: 126 Convex Bodies 7.3 Characterizat
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Page 141 and 142: 130 Convex Bodies Since the Riemann
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Page 145 and 146: 134 Convex Bodies Mean Width and th
Page 147 and 148: 136 Convex Bodies � χ(C ∩ mD)
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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
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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
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Page 179 and 180: 168 Convex Bodies Clearly, the inte
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Page 183 and 184: 172 Convex Bodies Proposition 9.2.
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Page 189 and 190: 178 Convex Bodies 9.3 Schwarz Symme
Page 191 and 192: 180 Convex Bodies Torsional Rigidit
Page 193 and 194: 182 Convex Bodies The proof of (3)
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Page 199 and 200: 188 Convex Bodies Both problems inf
Page 201 and 202: 190 Convex Bodies The Area Measure
Page 203 and 204: 192 Convex Bodies halfspace {z : un
Page 205 and 206: 194 Convex Bodies Let σm be the di
Page 207 and 208: 196 Convex Bodies Thus Minkowski’
Page 209 and 210: 198 Convex Bodies Suppose now that
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Page 219 and 220: 208 Convex Bodies Note that ui ∈
Page 223 and 224: 212 Convex Bodies the maximum of th
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Page 227 and 228: 216 Convex Bodies Prove analogous r
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Page 231 and 232: 220 Convex Bodies λC + x νC + z
Page 233 and 234: 222 Convex Bodies To show that (K +
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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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