Gruber P. Convex and Discrete Geometry

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16 Volume of Polytopes and Hilbert’s Third Problem 285 Since we have already proved the assertion (10) for m = 2, the induction assumption on m then shows that V (P) = V � (P ∩ H − ) ˙∪(P ∩ H + ) � = V (P ∩ H − ) + V (P ∩ H + ) = V � P1 ˙∪··· ˙∪Pj ˙∪(Pj+1 ∩ H − ) ˙∪··· ˙∪(Pk ∩ H − ) � + V � (Pj+1 ∩ H + ) ˙∪··· ˙∪(Pk ∩ H + ) ˙∪Pk+1 ˙∪··· � ˙∪Pm = V (P1) +···+V (Pj) + V (Pj+1 ∩ H − ) +···+V (Pk ∩ H − ) + V (Pj+1 ∩ H + ) +···+V (Pk ∩ H + ) + V (Pk+1) +···+V (Pm) = V (P1) +···+V (Pj) + V (Pj+1) +···+V (Pk) + V (Pk+1) +···+V (Pm). The induction on m is thus complete. Hence (10) holds for d and all m. This concludes the induction on d for (10). For the proof of the assertion (11) note first that V (P) = 0forP ∈ P with dim P ≤ d − 1. Hence V is simple. To see that it is a valuation, let P, Q ∈ P such that P ∪ Q ∈ P. The cases where at least one of P, Q, P ∩ Q is improper are easily dealt with by the simplicity of V and (10). Assume now that P, Q, P ∩ Q all are proper. Consider all hyperplanes through facets of P ∩ Q. These hyperplanes dissect P and Q into proper convex polytopes say. Then P ∩ Q, P1,...,Pm and P ∩ Q, Q1,...,Qn, P ∩ Q, P1,...,Pm, Q1,...,Qn form a dissection of P ∪ Q and it follows from (10) that V (P) + V (Q) = V (P ∩ Q) + V (P1) +···+V (Pm) + V (Q1) +···+V (Qn) + V (P ∩ Q) = V (P ∪ Q) + V (P ∩ Q). Thus V is a valuation, concluding the induction for (11). Assertion (12) easily follows from the definition of V by induction on d. If dim Z < d, the assertion (13) holds trivially. Assume now that dim Z = d. To speak more easily, call L vertical for the proof of (13). Let S be a parallel slab as in (13) and T a horizontal slab of width l. For the proof of (13) it is sufficient to show that (17) V (S ∩ Z) = V (T ∩ Z). If S and T are parallel, this is easy to see. Assume now that S and T are not parallel. By induction, (18) The contributions of the vertical facets of S ∩ Z and T ∩ Z to V (S ∩ Z) and V (T ∩ Z), respectively, are the same. Next, the non-vertical facets will be considered. Let v be a unit vector parallel to L such that F, F − lv and G, G − lv are the top and bottom facets of S ∩ Z and
286 Convex Polytopes T ∩ Z, respectively. Let ±u be the exterior normal unit vectors of S ∩ Z at the facets F, F − lv. Clearly, ±v are the exterior normal unit vectors of T ∩ Z at the facets G, G − lv. Choose p ∈ F and q ∈ G arbitrarily. Then (19) The contributions of the facets F, F − lv of S ∩ Z to V (S ∩ Z) and of the facets G, G − lv of T ∩ Z to V (T ∩ Z) are 1 � � l p · u v(F) + (p − lv) · (−u) v(F−lv) = u · vv(F), d d 1 � � l q · vv(G) + (q − lv) · (−v) v(G − lv) = d d v(G). Here, the translation invariance of v was used. Clearly, G = F ′ , where “ ′ ”is the orthogonal projection of the hyperplane HF = aff F onto the hyperplane HG = aff G. Since S, T are not parallel, HF, HG are also not parallel. Let w : P(HF) → R be defined by: w(Q) = v(Q ′ ) for Q ∈ P(HF). Since, by induction, v is a simple translation invariant, monotone valuation on P(HG), it is easy to see, that this also holds for w on P(HF). Thus, there are two simple, translation invariant valuations v and w on P(HF). The uniqueness theorem 16.1 then yields w(Q) = αv(Q) for Q ∈ P(HF), where α ≥ 0 is a suitable constant. To determine its value, choose a (d − 1)dimensional cube K in HF of edge-length 1 such that a (d − 2)-dimensional facet of K is contained in HF ∩ HG. Then K ′ is a box in HG of edge-lengths 1,...,1, u · v. Thus w(K ) = v(K ′ ) = u · v = αv(K ) = α, i.e. α = u · v. Here we have used property (12) for v. Hence By (19), this shows that v(G) = v(F ′ ) = αv(F) = u · vv(F). (20) The contributions of the facets F, F − lv of S ∩ Z to V (S ∩ Z) and of the facets G, G − lv of T ∩ Z to V (T ∩ Z) are the same. Having proved (18) and (20), the proof of (17) is complete, concluding the induction for (13). For the proof of (14), let P ∈ P and t ∈ E d .IfP is improper, V (P) = 0 = V (P + t) since V is simple by (11), and we are done. Assume now that P is proper. Let L be a line through o with direction t and call it vertical. Let H be the (d − 1)dimensional subspace of E d orthogonal to L. LetF1,...,Fm be the facets of P on the upper side and G1,...,Gn the facets on the lower side of P. The translate P + t of P may be obtained as follows. Dissect each facet Fi into pieces Fi1,...,Fin and each facet G j into pieces G1 j,...,Gmj such that
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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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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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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
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: 284 Convex Polytopes (12) The volum
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
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
Page 331 and 332: 320 Convex Polytopes � qP(t) = No
Page 333 and 334: 322 Convex Polytopes Now, take into
Page 335 and 336: 324 Convex Polytopes L � P + (n +
Page 337 and 338: 326 Convex Polytopes V (S) = V .Let
Page 339 and 340: 328 Convex Polytopes (7) ψ|H d = 0
Page 341 and 342: 330 Convex Polytopes Theorem 19.6.
Page 343 and 344: 332 Convex Polytopes (15) L(nP) = L
Page 345 and 346: 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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