Gruber P. Convex and Discrete Geometry

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(7) χ � C(Fi) � = 1fori = 1,...,m, 15 Combinatorial Theory of Convex Polytopes 269 settling (i). The assertion for i = 1 in (ii) is a consequence of (i). Assume next that 1 < i < m. Then (8) χ � C(F1 ∪···∪ Fi) � = χ � C(F1 ∪···∪ Fi−1) � + χ � C(Fi) � − χ � C((F1 ∪···∪Fi−1) ∩ Fi) � = χ � C(F1 ∪···∪ Fi−1) � + 1 − χ � C(G1 ∪···∪G j) � = χ � C(F1 ∪···∪ Fi−1) � + 1 − 1 = χ � C(F1 ∪···∪ Fi−1) � =···=χ � C(F1) � = 1 by the additivity of χ, where j < k, G1,...,Gk is a shelling of Fi and we have used induction on (ii) and (7). Assume next that i = m. Then χ � C(F1 ∪···∪ Fm) � = χ � C(F1 ∪···∪ Fm−1) � + χ � C(Fm) � − χ � � C(F1 ∪···∪Fm−1) ∩ Fm = 1 + 1 − � 1 − (−1) d−1� = 1 + (−1) d by the additivity of χ, (8), (7), noting that C((F1 ∪···∪Fm−1)∩ Fm) is the boundary complex of Fm, and induction on (ii). The proof of (ii) is complete. To show (iii), note that C(P) is obtained from C(F1 ∪···∪Fm) by adding P. Hence the definition of χ shows that χ � C(P) � = χ � C(F1 ∪···∪Fm) � + (−1) d = 1 − (−1) d + (−1) d = 1. This settles (iii). The induction and thus the proof of (6) is complete. ⊓⊔ Remark. It can be shown that the Euler polytope formula is the only linear relation which is satisfied by the f -vectors of all d-dimensional convex polytopes. The Dehn–Sommerville Equations and McMullen’s g-Theorem At present a characterization of all f -vectors of proper convex polytopes, i.e. a solution of Problem 15.1 for d > 3, seems to be out of reach. But for simplicial and simple convex polytopes this can be done. These are convex polytopes all facets of which are simplices, resp. convex polytopes, each vertex of which is incident with precisely d edges. We consider only the simplicial case. Dehn [251] (d = 5) and Sommerville [948] (general d) specified ⌊ d+1 2 ⌋ independent linear equations, including the Euler polytope formula (1), which are satisfied by the f -vectors of all proper simplicial convex polytopes. It can be shown that there is no linear relation independent of the Dehn–Sommerville equations which is satisfied by the f -vectors of all proper simplicial convex polytopes, see Grünbaum [453], Sect. 9.2. McMullen [707] stated in 1970 a characterization of the set of all f -vectors of proper simplicial convex polytopes in Ed ,theg-conjecture. This conjecture was confirmed by the efforts of Stanley [951], who used heavy algebraic machinery (necessity of McMullen’s conditions) and Billera and Lee [116] (sufficiency). See also McMullen [715].
270 Convex Polytopes The Lower and the Upper Bound Theorem It is a natural question to determine the best lower and upper bounds for fi(P), i = 1,...,d − 1, for all convex polytopes P with given n = f0(P) (or fk(P) for given k). The best lower bounds are attained by so-called stacked polytopes, as shown by Barnette [73, 74]. In particular, fd−1(P) ≥ (d − 1)n − (d + 1)(d − 2). The best upper bounds are attained by so-called neighbourly polytopes. This was shown by McMullen [705], thereby confirming Motzkin’s [758] upper bound conjecture. In particular, where d 2� ∗ i=0 fd−1(P) ≤ 2 ai = d 2� � � ∗ n − d − 1 + i , i i=0 � a0 +···+a d−1 2 a0 +···+a d−2 2 + 1 2 a d 2 for d odd, for d even. 15.3 Steinitz’ Polytope Representation Theorem for d = 3 The edge graph of a proper convex polytope P in E 3 is planar and 3-connected. To see the former property, project the 1-skeleton of P from an exterior point of P which is sufficiently close to a given relative interior point of a facet F of P onto a plane parallel to F on the far side of P. For the latter property, see Theorem 15.8 later. The proof that, conversely, a planar, 3-connected graph can be realized as the edge graph of a convex polytope in E 3 is due to Steinitz [963,965]. Grünbaum [453], Sect. 13.1, stated in 1967 that this is The most important and deepest known result on 3-polytopes ... So far, it has resisted extension to higher dimensions. Since the determination of the face lattices of 4-dimensional convex polytopes is NP-hard, almost surely there is no such extension. In spite of this, the Steinitz theorem led to a collection of non-trivial results of modern polytope theory centred around the following questions. Problem 15.2. Characterize, among all graphs or in a given family of graphs, those which can be realized as edge graphs of convex polytopes. Problem 15.3. Given the edge graph of a convex polytope, what can be said about the polytope? In this section we give a short proof of the Steinitz theorem using the Koebe– Brightwell–Scheinerman representation theorem 34.1 for planar graphs. Section 15.4 contains further contributions to these problems.
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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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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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Page 233 and 234: 222 Convex Bodies To show that (K +
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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
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Page 283 and 284: 272 Convex Polytopes A problem of S
Page 285 and 286: 274 Convex Polytopes The first cond
Page 287 and 288: 276 Convex Polytopes The existence
Page 289 and 290: 278 Convex Polytopes For d = 3 it f
Page 291 and 292: 280 Convex Polytopes 16 Volume of P
Page 293 and 294: 282 Convex Polytopes by (2). Hence
Page 295 and 296: 284 Convex Polytopes (12) The volum
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Page 299 and 300: 288 Convex Polytopes An Alternative
Page 301 and 302: 290 Convex Polytopes P clearly can
Page 303 and 304: 292 Convex Polytopes 17 Rigidity Ri
Page 305 and 306: 294 Convex Polytopes Fig. 17.2. Cau
Page 307 and 308: 296 Convex Polytopes Call the index
Page 309 and 310: 298 Convex Polytopes of V. Fori ∈
Page 311 and 312: 300 Convex Polytopes Thus (3) impli
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Page 315 and 316: 304 Convex Polytopes For other pert
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Page 319 and 320: 308 Convex Polytopes 18.3 The Isope
Page 321 and 322: 310 Convex Polytopes 19 Lattice Pol
Page 323 and 324: 312 Convex Polytopes Embed E d into
Page 325 and 326: 314 Convex Polytopes To see this, n
Page 327 and 328: 316 Convex Polytopes The Coefficien
Page 329 and 330: 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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