Gruber P. Convex and Discrete Geometry

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15 Combinatorial Theory of Convex Polytopes 261 4A(T ) + 4π = 2α + 2α + 2β + 2β + 2γ + 2γ, or A(T ) = α + β + γ − π. ⊓⊔ Proof of the polytope formula. We may assume that o ∈ int P. Projecting bd P from o radially onto S 2 yields a one-fold covering of S 2 by f convex spherical polygons. Apply the above lemma to each of these polygons. Note that the sum of the interior angles of such polygons at a common vertex is 2π and that each edge is an edge of precisely two of these polygons, and add. Then (1) A(S 2 ) = 4π = 2πv − 2πe + 2π f, or v − e + f = 2. Since each edge is incident with two vertices and each vertex with at least three edges, it follows that 3v ≤ 2e, similarly, 3 f ≤ 2e. By (1), Adding then yields that −2v =−2e + 2 f − 4, −2 f =−2e + 2v − 4. v ≤ 2 f − 4, f ≤ 2v − 4. ⊓⊔ Abstract Graphs, Graphs and Realizations Before proceeding to the graph version of Euler’s formula, some notation will be introduced. A (finite) abstract graph G consists of two sets, the set of vertices V = V(G) = {1,...,v} and the set of edges E = E(G) which consists of two-element subsets of V. If{u,w} is an edge, we write also uw for it and call w a neighbour of u and vice versa. If a vertex is contained in an edge or an edge contains a vertex then the vertex and the edge are incident. If two edges have a vertex in common, they are incident at this vertex. A path is a sequence of edges such that each edge is incident with the preceding edge at one vertex and with the next edge at the other vertex. We may write a path also as a sequence w1w2 ···wk of vertices. We say that it connects the first and the last vertex of this sequence. A path is a cycle if the first and the last vertex coincide. A cycle which consists of k edges is a k-cycle. G is connected if any two distinct of its vertices are connected by a path. If G has at least k + 1 vertices, it is kconnected if any two distinct vertices can be connected by k paths which, pairwise, have only these vertices in common. With some effort it can be shown that this is equivalent to the following. The graph which is obtained from G by deleting any set of k − 1 vertices and the edges incident with these, is still connected. More geometrically, by definition, a (finite) graph consists of a finite point set in some space, the set of vertices and, for each of a set of pairs of distinct vertices, of a continuous curve which connects these vertices and contains no vertex in its relative interior. These curves form the set of edges. The notions and notations which were defined earlier for abstract graphs are defined for graphs in the obvious way. Two graphs G, H, one or both of which may be abstract, are isomorphic if there is a
262 Convex Polytopes bijection between V(G) and V(H) which maps (endpoints of) edges onto (endpoints of) edges in both directions. A graph in E 2 (or C or C ∪{∞})isplanar if no two edges of it cross. More generally, an abstract graph or a graph is called planar if it is isomorphic to a planar graph in E 2 . The isomorphic image in E 2 then is a planar realization of the given, possibly abstract graph. There are simple criteria for planarity. Let G be a planar connected graph in E 2 . Omitting from E 2 the vertices and edges of G leaves a finite system of connected open sets, the countries of G, often called domains of G. Since G is connected, the bounded countries are simply connected. If G is considered as embedded into C ∪{∞}, also the unbounded country is simply connected. It is easy to see that each planar graph in E 2 is isomorphic to a planar graph in E 2 the edges of which are polygonal curves. The edge graph G(P) of a proper convex polytope P is the graph consisting of the vertices and the edges of P. IfdimP = 3, then G(P) is planar. For more information on geometric graph theory, see Nishizeki and Chiba [772], Bollobás [142], Mohar and Thomassen [748] and Felsner [332]. Euler’s Formula for Graphs in E 2 We will show the following result. Theorem 15.2. Let G be a connected planar graph in E 2 with v vertices, e edges and f countries. Then v − e + f = 2. Proof. Define the Euler characteristic χ(H) of a connected planar graph H in E 2 as the number of vertices minus the number of edges plus the number of countries of H, including the unbounded country. For simplicity, we assume that all graphs considered have polygonal curves as edges. The following proposition is well-known. (2) Let a graph K in E 2 arise from a connected planar graph H in E 2 by one of the following two operations: (i) Consider a polygonal curve connecting a vertex of H with a point in a country of H and which contains no vertex or point on an edge of H in its relative interior. Add this curve to the edges of H and its endpoint to the vertices. (ii) Consider a polygonal curve connecting two vertices of H and which contains no vertex or point on an edge of H in its relative interior. Add this curve to the edges of H. Then χ(K) = χ(H). Since G is connected, it can be constructed by finitely many operations as described in (2), beginning with a graph consisting of a single vertex which has Euler characteristic 2. Proposition (2) then shows that χ(G) = 2. ⊓⊔
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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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186 Convex Bodies Theorem 9.10. Let
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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 ∈
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
Page 229 and 230: 218 Convex Bodies Heuristic Princip
Page 231 and 232: 220 Convex Bodies λC + x νC + z
Page 233 and 234: 222 Convex Bodies To show that (K +
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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: 260 Convex Polytopes Euler’s Poly
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
Page 281 and 282: 270 Convex Polytopes The Lower and
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
Page 297 and 298: 286 Convex Polytopes T ∩ Z, respe
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
Page 313 and 314: 302 Convex Polytopes Proof. (i)⇒(
Page 315 and 316: 304 Convex Polytopes For other pert
Page 317 and 318: 306 Convex Polytopes where λ is a
Page 319 and 320: 308 Convex Polytopes 18.3 The Isope
Page 321 and 322: 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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