Gruber P. Convex and Discrete Geometry

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Metric Projection C pC (x) x pC (y) y S 4 Support and Separation 53 Hx Fig. 4.1. Metric projection is non-expansive Let C be a closed convex set in Ed . For each x ∈ Ed , there is a unique point pC(x) ∈ C closest to it. Since C is closed, the existence is obvious. To see the uniqueness, assume that there are points y, z ∈ C, y �= z, both having minimum distance from x. Then �y − x� =�z− x� and therefore � 1 2 (y + z) − x� < �y − x�, �z − x�, noting that y �= z. Since 1 2 (y + z) ∈ C by the convexity of C, this contradicts our choice of y, z. The mapping pC : x → pC(x) of Ed onto C, thus obtained, is the metric projection of Ed onto C with respect to the Euclidean norm. The following useful result is due to Busemann and Feller [183]. Lemma 4.1. Let C ⊆ E d be a closed convex set. Then the metric projection pC : E d → Cisnon-expansive, i.e. Hy �pC(x) − pC(y)� ≤�x − y� for x, y ∈ E d . Given a hyperplane H in E d ,letH + and H − denote the closed halfspaces determined by H. Wesay,H separates two sets, if one set is contained in H + and the other one in H − . Proof. Let x, y ∈ E d . We consider only the case where x, y �∈ C (Fig. 4.1). The other cases are treated similarly. If pC(x) = pC(y), we are done. Assume then that pC(x) �= pC(y).LetS be the slab orthogonal to the line segment [pC(x), pC(y)] ⊆ C and such that its boundary hyperplanes Hx and Hy contain pC(x) and pC(y), respectively. We claim that x and pC(y) are separated by Hx. Otherwise there is a point on [pC(x), pC(y)] and thus in C which is closer to x than pC(x), which is impossible. Similarly, y and pC(x) are separated by Hy. Taken together, this means that x and y are on different sides of the slab S. Hence �x − y� is at least equal to the width of S, that is �x − y� ≥�pC(x) − pC(y)�. ⊓⊔ Support Hyperplanes, Normal Vectors and Support Sets Let C ⊆ E d be closed and convex. A hyperplane H = HC(x) is a support hyperplane of C at a point x of the boundary bd C of C, ifx ∈ HC(x) and C is contained
54 Convex Bodies in one of the two closed halfspaces determined by H. In this case, we denote the halfspace containing C by H − , the other one by H + . H − is called a support halfspace of C at x. In general, we represent H in the form H ={z : u · z = u · x}, where u is a normal (unit) vector of H pointing into H + . Then H − ={z : u · z ≤ u · x} and H + ={z : u · z ≥ u · x}. u is an exterior normal (unit) vector of H, ofC or of bd C at x. Note that H may not be unique. The intersection C ∩ H is the support set of C determined by H or, with exterior normal (unit) vector u. In Theorems 1.2 and 2.3, it was shown that a convex function has affine support at each point in the interior of its domain of definition. The following theorem is the corresponding result for convex sets. Theorem 4.1. Let C ⊆ E d be a closed convex set. For each x ∈ bd C, there is a support hyperplane HC(x) of C at x, not necessarily unique. If C is compact, then for each vector u ∈ E d \{o}, there is a unique support hyperplane HC(u) of C with exterior normal vector u. Let S d−1 denote the Euclidean unit sphere in E d . Proof. First, the following will be shown. (1) Let y ∈ E d \C. Then the hyperplane H through pC(y) ∈ bd C, orthogonal to y − pC(y), supports C at pC(y). It is sufficient to show that H separates y and C. If this does not hold, there is a point z ∈ C which is not separated from y by H. Then the line segment [pC(y), z] ⊆C contains a point of C which is closer to y than pC(y). This contradicts the definition of pC(y) and thus concludes the proof of (1). Next we claim the following. (2) Let Hn ={z : un · z = xn · un} be support hyperplanes of C at the points xn ∈ bd C, n = 1, 2,... Assume that un → u ∈ E d \{o} and xn → x (∈ bd C) as n →∞. Then H ={z : u · z = u · x} is a support hyperplane of C at x. Clearly, x ∈ H. It is sufficient to show that C ⊆ H − .Letz ∈ C. Then un ·z ≤ un · xn for n = 1, 2,... Letting n →∞, we see that u · z ≤ u · x, or z ∈ H − , concluding the proof of (2). For the proof of the first assertion in the theorem, choose points yn ∈ E d \C, n = 1, 2,...,such that yn → x. By Lemma 4.1, xn = pC(yn)(∈ bd C) → x = pC(x). Proposition (1) shows that, for n = 1, 2,..., there is a support hyperplane of C at xn,sayHn ={z : un · z = un · xn}, where un ∈ S d−1 . By considering a subsequence and re-numbering, if necessary, we may suppose that un → u ∈ S d−1 ,say.An application of (2) then implies that H ={z : u · z = u · x} is a support hyperplane of C at x. To see the second assertion, note that the compactness of C implies that u · x = sup{u · z : z ∈ C} for a suitable x ∈ C. Clearly, x ∈ bd C and H ={z : u · z = u · x} is a support hyperplane of C (at x) with exterior normal vector u. ⊓⊔
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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
Page 13 and 14: 2 Convex Functions 1 Convex Functio
Page 15 and 16: 4 Convex Functions x epi f f C y Fi
Page 17 and 18: 6 Convex Functions Together these i
Page 19 and 20: 8 Convex Functions Theorem 1.4. Let
Page 21 and 22: 10 Convex Functions z = f (x) + f
Page 23 and 24: 12 Convex Functions 1.3 Convexity C
Page 25 and 26: 14 Convex Functions Proof (by affin
Page 27 and 28: 16 Convex Functions Minkowski’s I
Page 29 and 30: 18 Convex Functions (ii) Let x > 0.
Page 31 and 32: 20 Convex Functions 2 Convex Functi
Page 33 and 34: 22 Convex Functions | f (z) − f (
Page 35 and 36: 24 Convex Functions First-Order Dif
Page 37 and 38: 26 Convex Functions Remark. More pr
Page 39 and 40: 28 Convex Functions even more is tr
Page 41 and 42: 30 Convex Functions G(x). The proof
Page 43 and 44: 32 Convex Functions by (15), unifor
Page 45 and 46: 34 Convex Functions 2.4 A Stone-Wei
Page 47 and 48: 36 Convex Functions x x Fig. 2.1. S
Page 49 and 50: 38 Convex Functions F is convex. Us
Page 51 and 52: 40 Convex Bodies results, character
Page 53 and 54: 42 Convex Bodies Convex Hulls Given
Page 55 and 56: 44 Convex Bodies The next result is
Page 57 and 58: 46 Convex Bodies C ∩ L ⊥ clearl
Page 59 and 60: 48 Convex Bodies Suppose not. Then,
Page 61 and 62: 50 Convex Bodies The Centrepoint of
Page 63: 52 Convex Bodies Since G is open, w
Page 67 and 68: 56 Convex Bodies or geometric infor
Page 69 and 70: 58 Convex Bodies v HC (u, h(u)) (v,
Page 71 and 72: 60 Convex Bodies C D Fig. 4.3. Sepa
Page 73 and 74: 62 Convex Bodies For the proof of t
Page 75 and 76: 64 Convex Bodies in contradiction t
Page 77 and 78: 66 Convex Bodies (6) R(t) ⊆ E d i
Page 79 and 80: 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
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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
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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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