Gruber P. Convex and Discrete Geometry

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12 Special Convex Bodies 219 Characterizations of Simplices of Choquet and Rogers–Shephard Choquet [208] ((i)⇔(ii)) and Rogers and Shephard [852] ((i)⇔(iii)) proved the following seminal result. Theorem 12.1. Let C ∈ C. Then the following statements are equivalent: (i) Cisasimplex. (ii) For any λ, µ ≥ 0 and x, y ∈ E d with (λC + x) ∩ (µC + y) �= ∅, there are ν ≥ 0 and z ∈ E d such that (λC + x) ∩ (µC + y) = νC + z. (iii) For any x, y ∈ E d with (C + x) ∩ (C + y) �= ∅, there are ν ≥ 0 and z ∈ E d such that (C + x) ∩ (C + y) = νC + z. An exposed point of a convex body C is a point p ∈ C, such that there is a support hyperplane H of C with H ∩ C ={p}. Proof. The implications (i)⇒(ii) and (ii)⇒(iii) are easy to prove (consider the simplex {x : 0 ≤ xi, x1 +···+xd ≤ 1}), or trivial. Thus it is sufficient to show that (iii)⇒(i) We may suppose that dim C = d. The proof is by induction on d. For d = 0, 1 each convex body and thus, in particular, C is a simplex. Assume now that d > 1 and that the implication (iii)⇒(i) holds for dimensions 0, 1,...,d − 1. C contains an exposed boundary point, take for example a point where a circumscribed sphere touches C. After a translation, if necessary, we may suppose that this point is the origin o. Choose a support hyperplane H of C with C ∩ H ={o}. Since C is proper, the smooth points are dense in bd C, see Theorem 5.1. Thus we may choose a smooth point p ∈ bd C such that the open line segment with endpoints o, p is contained in int C. In order to see that (1) C ∩ (C − λp) = (1 − λ)C for 0
220 Convex Bodies λC + x νC + z µC + y Fig. 12.1. Choquet simplex w ∈ C ∩ (C − λp) = (1 − λ)C, (see (1)) and w = (1 − λ)x, we have that x ∈ C. Thus (cone C) ∩ K − ⊆ C. The proof of (2) is complete. For each vector x parallel to the hyperplane K and such that F ∩ (F + x) �= ∅, the set F ∩ (F + x) is a face of C ∩ (C + x), and C ∩ (C + x) is homothetic to C by (iii). Hence F ∩ (F + x) is homothetic to F. The induction hypothesis then shows that F is a simplex. Since o �∈ K , Proposition (2) finally implies that C is also a simplex, concluding the proof of (i). ⊓⊔ Remark. A compact convex set C in an infinite-dimensional topological vector space which satisfies property (iii) is called a Choquet simplex (see Fig. 12.1). Choquet simplices are basic in Choquet theory and thus in measure theory in infinite dimensions. Topological Vector Lattices in Finite Dimensions Let V be a real vector space and � a(partial) ordering of V in the following sense: x � x for x ∈ V x � y, y � x ⇒ x = y for x, y ∈ V x � y, y � z ⇒ x � z for x, y, z ∈ V 〈V, �〉 is an ordered vector space if, in addition, the ordering is compatible with the operations in V , i.e. x � y ⇒ x + z � y + z for x, y, z ∈ V x � y ⇒ λx � λy for x, y ∈ V, λ≥ 0 Then K ={x ∈ V : o � x} is a convex cone with apex o such that K ∩ (−K ) = {o}, that is, K is pointed. K is called the positive cone (see Fig. 12.2) of 〈V, �〉. Conversely, if K is a pointed convex cone in V with apex o, then the definition (3) x � y if y − x ∈ K or, equivalently, y ∈ K + x for x, y ∈ V makes V into an ordered vector space with positive cone K . The ordered vector space 〈V, �〉 is a vector lattice, if for any x, y ∈ V there is a greatest lower bound x ∧ y and a smallest upper bound x ∨ y of the set {x, y} in V , that is,
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Grundlehren der mathematischen Wiss
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Peter M. Gruber Institute of Discre
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VI Preface theory of finite groups
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Contents Preface ..................
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Contents XI 12 Special Convex Bodie
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Contents XIII 29 Packing of Balls a
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2 Convex Functions 1 Convex Functio
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4 Convex Functions x epi f f C y Fi
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6 Convex Functions Together these i
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8 Convex Functions Theorem 1.4. Let
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10 Convex Functions z = f (x) + f
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12 Convex Functions 1.3 Convexity C
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14 Convex Functions Proof (by affin
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16 Convex Functions Minkowski’s I
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18 Convex Functions (ii) Let x > 0.
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20 Convex Functions 2 Convex Functi
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22 Convex Functions | f (z) − f (
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24 Convex Functions First-Order Dif
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26 Convex Functions Remark. More pr
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28 Convex Functions even more is tr
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30 Convex Functions G(x). The proof
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32 Convex Functions by (15), unifor
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34 Convex Functions 2.4 A Stone-Wei
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36 Convex Functions x x Fig. 2.1. S
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38 Convex Functions F is convex. Us
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40 Convex Bodies results, character
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42 Convex Bodies Convex Hulls Given
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44 Convex Bodies The next result is
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46 Convex Bodies C ∩ L ⊥ clearl
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48 Convex Bodies Suppose not. Then,
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50 Convex Bodies The Centrepoint of
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52 Convex Bodies Since G is open, w
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54 Convex Bodies in one of the two
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56 Convex Bodies or geometric infor
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58 Convex Bodies v HC (u, h(u)) (v,
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60 Convex Bodies C D Fig. 4.3. Sepa
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62 Convex Bodies For the proof of t
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64 Convex Bodies in contradiction t
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66 Convex Bodies (6) R(t) ⊆ E d i
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68 Convex Bodies 5 The Boundary of
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70 Convex Bodies for all y ∈ C an
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72 Convex Bodies the existence of s
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74 Convex Bodies where r(·) is a s
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76 Convex Bodies Hence x ∈ conv e
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78 Convex Bodies N = � � X, O Y
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80 Convex Bodies Further topics of
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82 Convex Bodies We are now ready t
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84 Convex Bodies similarly, D j is
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86 Convex Bodies For the proof of t
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88 Convex Bodies The definition of
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90 Convex Bodies If ei ∈ Pi is no
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92 Convex Bodies Let v(·) denote (
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94 Convex Bodies The present sectio
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96 Convex Bodies Proposition 6.5. L
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98 Convex Bodies Thus Second, Minko
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100 Convex Bodies The Valuation Pro
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102 Convex Bodies Minkowski’s the
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104 Convex Bodies To remedy the sit
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106 Convex Bodies Proof. Applying S
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108 Convex Bodies Third, let C be a
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110 Convex Bodies A Problem of Sant
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112 Convex Bodies sets of S. Ifφ c
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114 Convex Bodies = � (−1) |J|
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116 Convex Bodies If P has dimensio
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118 Convex Bodies Since, by Volland
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120 Convex Bodies that is, V is sim
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122 Convex Bodies Let C, D ∈ C su
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124 Convex Bodies P Q . . . . Fig.
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126 Convex Bodies 7.3 Characterizat
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128 Convex Bodies and the Bi have p
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130 Convex Bodies Since the Riemann
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132 Convex Bodies Seventh, o v3 v2
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134 Convex Bodies Mean Width and th
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136 Convex Bodies � χ(C ∩ mD)
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138 Convex Bodies In the following,
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140 Convex Bodies 7.5 Hadwiger’s
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142 Convex Bodies 8.1 The Classical
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144 Convex Bodies C ∩ H(tC ) C D
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146 Convex Bodies Theorem 8.4. Let
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148 Convex Bodies In the following
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150 Convex Bodies Integration impli
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152 Convex Bodies As a consequence
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154 Convex Bodies Proof. � 1 2 w(
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156 Convex Bodies Let ϱ>0 be the m
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158 Convex Bodies Wulff’s Theorem
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160 Convex Bodies Third, by (8) and
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162 Convex Bodies Since the measure
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164 Convex Bodies An application of
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166 Convex Bodies Since f has Lipsc
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Page 181 and 182: 170 Convex Bodies Note that �st :
Page 183 and 184: 172 Convex Bodies Proposition 9.2.
Page 185 and 186: 174 Convex Bodies Proof. Let ε>0.
Page 187 and 188: 176 Convex Bodies The Brunn-Minkows
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)
Page 195 and 196: 184 Convex Bodies � �� 2 2
Page 197 and 198: 186 Convex Bodies Theorem 9.10. Let
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
Page 211 and 212: 200 Convex Bodies vector of St at x
Page 213 and 214: 202 Convex Bodies Outer Billiards F
Page 215 and 216: 204 Convex Bodies For more informat
Page 217 and 218: 206 Convex Bodies with P. Thus, it
Page 219 and 220: 208 Convex Bodies Note that ui ∈
Page 221 and 222: 210 Convex Bodies Theorem 11.4. Let
Page 223 and 224: 212 Convex Bodies the maximum of th
Page 225 and 226: 214 Convex Bodies (20) mni ≥ 1 λ
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 +
Page 235 and 236: 224 Convex Bodies The integral here
Page 237 and 238: 226 Convex Bodies In the following
Page 239 and 240: 228 Convex Bodies The first such re
Page 241 and 242: 230 Convex Bodies v o Fig. 12.3. Sy
Page 243 and 244: 232 Convex Bodies What Does the Bou
Page 245 and 246: 234 Convex Bodies Corollary 13.1. L
Page 247 and 248: 236 Convex Bodies Hence ν = 0 and
Page 249 and 250: 238 Convex Bodies geometry, see Sei
Page 251 and 252: 240 Convex Bodies We distinguish th
Page 253 and 254: 242 Convex Bodies by Proposition 13
Page 255 and 256: 244 Convex Polytopes The material i
Page 257 and 258: 246 Convex Polytopes V-Polytopes an
Page 259 and 260: 248 Convex Polytopes Convex cones w
Page 261 and 262: 250 Convex Polytopes Generalized Co
Page 263 and 264: 252 Convex Polytopes d 2 inequaliti
Page 265 and 266: 254 Convex Polytopes The Face Latti
Page 267 and 268: 256 Convex Polytopes Then (6) impli
Page 269 and 270: 258 Convex Polytopes P in F. For an
Page 271 and 272: 260 Convex Polytopes Euler’s Poly
Page 273 and 274: 262 Convex Polytopes bijection betw
Page 275 and 276: 264 Convex Polytopes The Converse o
Page 277 and 278: 266 Convex Polytopes (geometric) fa
Page 279 and 280: 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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