Gruber P. Convex and Discrete Geometry

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Mahler’s Inequality 9 Symmetrization 177 The following inequality of Mahler [679] is a useful tool for successive minima, see Theorem 23.2: Theorem 9.6. Let C ∈ Cp be o-symmetric. Then 4 d d ! ≤ V (C)V (C∗ ). Proof. Since a non-singular linear transformation does not change the product V (C)V (C ∗ ), we may assume the following. The o-symmetric cross-polytope O ={x :|x1|+···+|xd| ≤1} is inscribed in C and has maximum volume among all such cross-polytopes. Since O has maximum volume, C is contained in the cube K ={x :|xi| ≤1}. Thus O ⊆ C ⊆ K . Polarity then yields K ∗ ⊆ C ∗ ⊆ O ∗ .Note that K ∗ = O. Hence (2 d ) 2 (d !) 2 = V (O)2 = V (O)V (K ∗ ) ≤ V (C)V (C ∗ ), concluding the proof. ⊓⊔ A much stronger version of this simple result was conjectured also by Mahler: Conjecture 9.1. Let C ∈ Cp be o-symmetric. Then 4 d d ! ≤ V (C)V (C∗ ). Remark. It is plausible that the product V (C)V (C ∗ ) is minimum if C is the cube {x :|xi| ≤1}. Then C ∗ is the cross-polytope {x :|x1|+···+|xd| ≤1} and we have, V (C)V (C ∗ ) = 4 d /d !. This seems to have led Mahler to the above conjecture. Mahler proved it for d = 2. While the conjecture is open for d ≥ 3, substantial progress has been achieved by Bourgain and Milman [161] who proved that there is an absolute constant α such that α d d d ≤ V (C)V (C∗ ). A simple proof of a slightly weaker result is due to Kuperberg [622]. A refined version of the estimate of Bourgain and Milman was given by Kuperberg [624]. For special convex polytopes a proof of the conjecture is due to Lopez and Reisner [663]. In [623] Kuperberg stated a conjecture related to Mahler’s conjecture. For more information we refer to Lindenstrauss and Milman [660].
178 Convex Bodies 9.3 Schwarz Symmetrization and Rearrangement of Functions A relative of Steiner’s symmetrization is Schwarz’s symmetrization. It was introduced by Schwarz [922] as a tool for the solution of the geometric isoperimetric problem. While of interest in convex geometry, its real importance is in the context of isoperimetric inequalities of mathematical physics, where a version of it, the spherical rearrangement of functions, is an indispensable tool. In this section Schwarz symmetrization of convex bodies and spherically symmetric rearrangement of real functions are treated. Without proof we state a result on rearrangements. Applications are dealt with in the following sections. For references to the literature on rearrangement of functions and its applications to isoperimetric inequalities of mathematical physics and partial differential equations, consult the books of Bandle [64] and Kawohl [569] and the survey of Talenti [987]. Unfortunately, a modern treatment of this important topic with detailed proofs, still seems to be missing. Schwarz Symmetrization Schwarz symmetrization or Schwarz rounding of a convex body C with respect to a given line L is defined as follows. For each hyperplane H orthogonal to L and which meets C, replace C ∩ H by the (d − 1)-dimensional ball in H with centre at H ∩ L and (d − 1)-dimensional volume equal to that of C ∩ H. The union of all balls thus obtained is then the Schwarz symmetrization sch C = schL C of C with respect to L. The version 8.4 of the Brunn–Minkowski theorem implies that sch C ∈ C. Refining the argument that led to Corollary 9.1, shows that there are hyperplanes H1, H2,...,all containing L, such that st Hn ···st H1C → schLC. As a consequence, many properties of Steiner symmetrization also hold for Schwarz symmetrization; in particular, the properties listed in Proposition 9.1. Rearrangement of Functions It is clear that Schwarz symmetrization extends to non-convex sets (see Fig. 9.3). Given a Borel function f : E d → R, we apply Schwarz symmetrization to the set B ={(x, t) : f (x) ≥ t} ⊆E d × R with respect to the line {o}×R: Let B(t) ={(x, t) : f (x) ≥ t} ⊆E d ×{t} for t ∈ R. Since f , and thus B, is Borel, each set B(t) is also Borel. If B(t) has infinite measure, let B r (t) = E d ×{t}; otherwise let B r (t) be the ball in E d ×{t} with centre at (o, t) and measure equal to that of B(t). Therearrangement f r : E d → R of f then is defined by f r (x) = sup{t : (x, t) ∈ B r (t)} for x ∈ E d .
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Grundlehren der mathematischen Wiss
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Peter M. Gruber Institute of Discre
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VI Preface theory of finite groups
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Contents Preface ..................
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Contents XI 12 Special Convex Bodie
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Contents XIII 29 Packing of Balls a
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2 Convex Functions 1 Convex Functio
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4 Convex Functions x epi f f C y Fi
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6 Convex Functions Together these i
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8 Convex Functions Theorem 1.4. Let
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10 Convex Functions z = f (x) + f
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12 Convex Functions 1.3 Convexity C
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14 Convex Functions Proof (by affin
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16 Convex Functions Minkowski’s I
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18 Convex Functions (ii) Let x > 0.
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20 Convex Functions 2 Convex Functi
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22 Convex Functions | f (z) − f (
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24 Convex Functions First-Order Dif
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26 Convex Functions Remark. More pr
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28 Convex Functions even more is tr
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30 Convex Functions G(x). The proof
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32 Convex Functions by (15), unifor
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34 Convex Functions 2.4 A Stone-Wei
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36 Convex Functions x x Fig. 2.1. S
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38 Convex Functions F is convex. Us
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40 Convex Bodies results, character
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42 Convex Bodies Convex Hulls Given
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44 Convex Bodies The next result is
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46 Convex Bodies C ∩ L ⊥ clearl
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48 Convex Bodies Suppose not. Then,
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50 Convex Bodies The Centrepoint of
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52 Convex Bodies Since G is open, w
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54 Convex Bodies in one of the two
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56 Convex Bodies or geometric infor
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58 Convex Bodies v HC (u, h(u)) (v,
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60 Convex Bodies C D Fig. 4.3. Sepa
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62 Convex Bodies For the proof of t
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64 Convex Bodies in contradiction t
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66 Convex Bodies (6) R(t) ⊆ E d i
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68 Convex Bodies 5 The Boundary of
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70 Convex Bodies for all y ∈ C an
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72 Convex Bodies the existence of s
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74 Convex Bodies where r(·) is a s
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76 Convex Bodies Hence x ∈ conv e
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78 Convex Bodies N = � � X, O Y
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80 Convex Bodies Further topics of
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82 Convex Bodies We are now ready t
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84 Convex Bodies similarly, D j is
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86 Convex Bodies For the proof of t
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88 Convex Bodies The definition of
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90 Convex Bodies If ei ∈ Pi is no
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92 Convex Bodies Let v(·) denote (
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94 Convex Bodies The present sectio
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96 Convex Bodies Proposition 6.5. L
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98 Convex Bodies Thus Second, Minko
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100 Convex Bodies The Valuation Pro
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102 Convex Bodies Minkowski’s the
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104 Convex Bodies To remedy the sit
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106 Convex Bodies Proof. Applying S
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108 Convex Bodies Third, let C be a
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110 Convex Bodies A Problem of Sant
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112 Convex Bodies sets of S. Ifφ c
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114 Convex Bodies = � (−1) |J|
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116 Convex Bodies If P has dimensio
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118 Convex Bodies Since, by Volland
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120 Convex Bodies that is, V is sim
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122 Convex Bodies Let C, D ∈ C su
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124 Convex Bodies P Q . . . . Fig.
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)
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
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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 183 and 184: 172 Convex Bodies Proposition 9.2.
Page 185 and 186: 174 Convex Bodies Proof. Let ε>0.
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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
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Page 227 and 228: 216 Convex Bodies Prove analogous r
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Page 233 and 234: 222 Convex Bodies To show that (K +
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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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