Gruber P. Convex and Discrete Geometry

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(5) � C v(x) dx = lim � n→∞ C vn(x) dx = lim n→∞ αvn(o) = αv(o) 12 Special Convex Bodies 225 by Proposition (3). For the notions of stochastic convergence and uniform integrability and for the first equality in (5) see, e.g. Bauer [82], Sects. 20, 21. Finally, let a harmonic function w be defined by w(x) = 1 +�p�v(x) for x ∈ E 3 \{p} and let B be the ball �p�B 3 . Since o ∈ int C by (2), our choice of p shows that B ⊆ C. Thus (6) αw(o) = α � 1 +�p�v(o) � � = 0 = � � � 1 +�p�v(x) dx = w(x) dx by the definitions of w and v, (3) applied to u = 1 and Proposition (5). The mean value property of harmonic functions implies that � (7) w(x) dx = γw(o) = 0, B where γ > 0 is a suitable multiplicative constant, see Wermer [1018], Appendix. From (6) and (7) we conclude that � � � � 0 = w(x) dx = w(x) dx + w(x) dx = w(x) dx. C B Since w(x) > 0forx ∈ C\B, it follows that C\B has measure 0. Noting that C = cl int C by assumption, this means that C\B =∅,orC ⊆ B. The reverse inclusion being obvious, C = B follows. ⊓⊔ C C\B C\B 12.3 Blaschke’s Characterization of Ellipsoids and Its Applications Ellipsoids and Euclidean spaces play an important role in many branches of mathematics. Among these are convex and differential geometry, the local theory of normed spaces, functional analysis, approximation, operator and potential theory, dynamical systems, combinatorial optimization and mechanics. The first characterizations of ellipsoids in convex geometry go back to Brunn and Blaschke, and at present there is still interest for these characterizations. One may distinguish between characterizations based on: Affine and projective transformations Sections Projections and illuminations Extremal properties Other geometric and analytic properties C
226 Convex Bodies In the following we consider a characterization of ellipsoids due to Blaschke [123,124] by the property that the shadow boundaries under parallel illumination all are planar. As a consequence, two characterizations of Euclidean spaces are given. A remark concerns geometric stability problems. Finally, we define Radon norms. For more information we refer to the books and surveys of Bonnesen and Fenchel [149], Day [248], Laugwitz [630], Gruber and Höbinger [446], Petty [798], Amir [27], Istrǎtescu [537], Heil and Martini [488], Lindenstrauss and Milman [660], Li, Simon and Zhao [655], Thompson [994], Martini [692], and Deutsch [263]. A Version of Blaschke’s Characterization of Ellipsoids Among all characterizations of ellipsoids in convex geometry, Blaschke’s [123, 124] characterization mentioned above has the largest number of applications. In some of these the following slightly refined version due to Marchaud [689] is used. Theorem 12.4. Let C be a proper convex body in E d , d ≥ 3. Then the following statements are equivalent: (i) C is an ellipsoid. (ii) For each line L through o there is a hyperplane H such that (1) C + L = C ∩ H + L. In other words, the shadow boundary bd(C + L)∩C of C under illumination parallel to L contains the “planar curve” bd(C + L) ∩ H. Before beginning with the proof we state two auxiliary results, the proofs of which are left to the reader but take some effort. Lemma 12.1. Let C ∈ Cp and k ∈{2,...,d}. Then the following assertions are equivalent: (i) C is an ellipsoid. (ii) For each k-dimensional plane H which meets int C, the intersection C ∩ Hisan ellipsoid. Lemma 12.2. Let D ∈ Cp(E 2 ) and w ∈ int D such that for any pair of parallel support lines S, T of D there are points s ∈ D ∩ S, t ∈ D ∩ T with w ∈[s, t]. Then w is the centre of C. Proof of the Theorem. The implication (i)⇒(ii) is easy. We show only that (ii)⇒(i) A simple argument implies that the intersection of C with any 3dimensional plane which meets int C also has property (ii). If the implication (ii)⇒(i) holds in case d = 3, the first lemma implies that C is an ellipsoid. It is thus sufficient to consider the case d = 3. Our first proposition is as follows: (2) Let U be a support plane of C. Then C ∩ U is not a line segment.
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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.
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
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Page 219 and 220: 208 Convex Bodies Note that ui ∈
Page 221 and 222: 210 Convex Bodies Theorem 11.4. Let
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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 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
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
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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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