Gruber P. Convex and Discrete Geometry

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C Steiner Symmetrization Fig. 9.1. Steiner symmetrization H 9 Symmetrization 169 Let C be a convex body and H a hyperplane in E d .TheSteiner symmetral st C = stH C of C (see Fig. 9.1) with respect to H is defined as follows: For each straight line L orthogonal to H and such that C ∩ L �= ∅, shift the line segment C ∩ L along L until its midpoint is in H. The union of all line segments thus obtained is st C. Clearly, st C is symmetric with respect to (mirror) reflection in H. According to Danilova [236], L’Huillier [654] and an anonymous author, possibly Gergonne [369], anticipated Steiner symmetrization in a vague form. Basic Properties of Steiner Symmetrization We first collect a series of simple results on Steiner symmetrization. Given a convex body C, theinradius r(C) and the circumradius R(C) are the maximum radius of a (solid Euclidean) ball contained in C and the minimum radius of a ball containing C, respectively. Proposition 9.1. Steiner symmetrization of convex bodies with respect to a given hyperplane H has the following properties: st H C (i) st C ∈ C for C ∈ C (ii) st λC = λ st C (up to translations) for λ ≥ 0, C ∈ C (iii) st(C + D) ⊇ st C + st D (up to translations) for C, D ∈ C (iv) st C ⊆ st DforC, D ∈ C, C ⊆ D, (v) st : Cp → Cp is continuous (vi) V (st C) = V (C) for C ∈ C (vii) S(st C) ≤ S(C) for C ∈ C (viii) diam st C ≤ diam CforC∈ C (ix) r(st C) ≥ r(C), R(st C) ≤ R(C) for C ∈ C
170 Convex Bodies Note that �st : �C →�� C is not continuous. Consider, for example, the sequence of line 1n segments o, , 1 ∈ C(E2 ), n = 1, 2,... Clearly, � � �� 1 o, , 1 →[o,(0, 1)] as n →∞, n but for Steiner symmetrization in the first coordinate axis we have that � � �� 1 1 st o, , 1 = o, , 0 →{o} �= 0, − n n 1 � � , 0, 2 1 �� = st [o,(0, 1)] . 2 Proof. We may assume that o ∈ H. LetL be the line through o orthogonal to the hyperplane H. (i) The proof that st C is compact is left to the reader. To show that st C is convex, let x, y ∈ st C. Consider the convex trapezoid T = conv �� C ∩ (L + x) � ∪ � C ∩ (L + y) �� ⊆ C. Since x, y ∈ st T and since st T is also a convex (see Fig. 9.2) trapezoid and is contained in st C, it follows that [x, y] ⊆st T ⊆ st C. (ii) Trivial. (iii) Let x ∈ st C, y ∈ st D or, equivalently, x = h + l, y = k + m, where h, k ∈ H and l, m ∈ L are such that �l� ≤ 1 2 length � C ∩ (L + x) � and �m� ≤ 1 2 length � D ∩ (L + y) � . Then x + y = (h + k) + (l + m) where h + k ∈ H (note that o ∈ H) and l + m ∈ L (note that o ∈ L), where �l + m� ≤�l�+�m� ≤ 1� � � � �� length C ∩ (L + x) + length D ∩ (L + y) 2 = 1 2 length � C ∩ (L + x) + D ∩ (L + y) � ≤ 1 2 length � (C + D) ∩ (L + x + y) � . Hence x + y ∈ st(C + D). T H y x st H T Fig. 9.2. Steiner symmetrization preserves convexity
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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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Page 131 and 132: 120 Convex Bodies that is, V is sim
Page 133 and 134: 122 Convex Bodies Let C, D ∈ C su
Page 135 and 136: 124 Convex Bodies P Q . . . . Fig.
Page 137 and 138: 126 Convex Bodies 7.3 Characterizat
Page 139 and 140: 128 Convex Bodies and the Bi have p
Page 141 and 142: 130 Convex Bodies Since the Riemann
Page 143 and 144: 132 Convex Bodies Seventh, o v3 v2
Page 145 and 146: 134 Convex Bodies Mean Width and th
Page 147 and 148: 136 Convex Bodies � χ(C ∩ mD)
Page 149 and 150: 138 Convex Bodies In the following,
Page 151 and 152: 140 Convex Bodies 7.5 Hadwiger’s
Page 153 and 154: 142 Convex Bodies 8.1 The Classical
Page 155 and 156: 144 Convex Bodies C ∩ H(tC ) C D
Page 157 and 158: 146 Convex Bodies Theorem 8.4. Let
Page 159 and 160: 148 Convex Bodies In the following
Page 161 and 162: 150 Convex Bodies Integration impli
Page 163 and 164: 152 Convex Bodies As a consequence
Page 165 and 166: 154 Convex Bodies Proof. � 1 2 w(
Page 167 and 168: 156 Convex Bodies Let ϱ>0 be the m
Page 169 and 170: 158 Convex Bodies Wulff’s Theorem
Page 171 and 172: 160 Convex Bodies Third, by (8) and
Page 173 and 174: 162 Convex Bodies Since the measure
Page 175 and 176: 164 Convex Bodies An application of
Page 177 and 178: 166 Convex Bodies Since f has Lipsc
Page 179: 168 Convex Bodies Clearly, the inte
Page 183 and 184: 172 Convex Bodies Proposition 9.2.
Page 185 and 186: 174 Convex Bodies Proof. Let ε>0.
Page 187 and 188: 176 Convex Bodies The Brunn-Minkows
Page 189 and 190: 178 Convex Bodies 9.3 Schwarz Symme
Page 191 and 192: 180 Convex Bodies Torsional Rigidit
Page 193 and 194: 182 Convex Bodies The proof of (3)
Page 195 and 196: 184 Convex Bodies � �� 2 2
Page 199 and 200: 188 Convex Bodies Both problems inf
Page 201 and 202: 190 Convex Bodies The Area Measure
Page 203 and 204: 192 Convex Bodies halfspace {z : un
Page 205 and 206: 194 Convex Bodies Let σm be the di
Page 207 and 208: 196 Convex Bodies Thus Minkowski’
Page 209 and 210: 198 Convex Bodies Suppose now that
Page 211 and 212: 200 Convex Bodies vector of St at x
Page 213 and 214: 202 Convex Bodies Outer Billiards F
Page 215 and 216: 204 Convex Bodies For more informat
Page 217 and 218: 206 Convex Bodies with P. Thus, it
Page 219 and 220: 208 Convex Bodies Note that ui ∈
Page 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
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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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234 Convex Bodies Corollary 13.1. L
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236 Convex Bodies Hence ν = 0 and
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238 Convex Bodies geometry, see Sei
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240 Convex Bodies We distinguish th
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242 Convex Bodies by Proposition 13
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244 Convex Polytopes The material i
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246 Convex Polytopes V-Polytopes an
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248 Convex Polytopes Convex cones w
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250 Convex Polytopes Generalized Co
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252 Convex Polytopes d 2 inequaliti
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254 Convex Polytopes The Face Latti
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256 Convex Polytopes Then (6) impli
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258 Convex Polytopes P in F. For an
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260 Convex Polytopes Euler’s Poly
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262 Convex Polytopes bijection betw
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264 Convex Polytopes The Converse o
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266 Convex Polytopes (geometric) fa
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268 Convex Polytopes consists preci
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270 Convex Polytopes The Lower and
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272 Convex Polytopes A problem of S
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274 Convex Polytopes The first cond
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276 Convex Polytopes The existence
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278 Convex Polytopes For d = 3 it f
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280 Convex Polytopes 16 Volume of P
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282 Convex Polytopes by (2). Hence
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284 Convex Polytopes (12) The volum
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286 Convex Polytopes T ∩ Z, respe
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288 Convex Polytopes An Alternative
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290 Convex Polytopes P clearly can
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292 Convex Polytopes 17 Rigidity Ri
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294 Convex Polytopes Fig. 17.2. Cau
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296 Convex Polytopes Call the index
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298 Convex Polytopes of V. Fori ∈
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300 Convex Polytopes Thus (3) impli
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302 Convex Polytopes Proof. (i)⇒(
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304 Convex Polytopes For other pert
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306 Convex Polytopes where λ is a
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308 Convex Polytopes 18.3 The Isope
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310 Convex Polytopes 19 Lattice Pol
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312 Convex Polytopes Embed E d into
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314 Convex Polytopes To see this, n
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316 Convex Polytopes The Coefficien
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318 Convex Polytopes If the triangl
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320 Convex Polytopes � qP(t) = No
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322 Convex Polytopes Now, take into
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324 Convex Polytopes L � P + (n +
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326 Convex Polytopes V (S) = V .Let
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328 Convex Polytopes (7) ψ|H d = 0
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330 Convex Polytopes Theorem 19.6.
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332 Convex Polytopes (15) L(nP) = L
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334 Convex Polytopes have to be sin
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336 Convex Polytopes economics, von
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338 Convex Polytopes Existence of S
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340 Convex Polytopes Description of
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342 Convex Polytopes (6) u piai xq
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344 Convex Polytopes For x ∈ Bd
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346 Convex Polytopes Rational Polyh
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348 Convex Polytopes Totally Dual I
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350 Convex Polytopes of the vectors
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Geometry of Numbers and Aspects of
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Extension to Crystallographic Group
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Relations Between Different Bases 2
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21 Lattices 359 triangle conv{o,(u,
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By definition of bi+1, Thus (6) yie
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21 Lattices 363 A similar, slightly
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21.4 Polar Lattices 21 Lattices 365
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The First Fundamental Theorem 22 Mi
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22 Minkowski’s First Fundamental
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If 22 Minkowski’s First Fundament
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22 Minkowski’s First Fundamental
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Proof. Consider the linear forms l1
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Proof. It is sufficient to consider
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Thus V (C) ≥ V (O) = 2d � �
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Given C and L, the problem arises t
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23 Successive Minima 383 as require
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24 The Minkowski-Hlawka Theorem 24
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(6) (V (J) ≈) � n�= 0 (7) J
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24 The Minkowski-Hlawka Theorem 389
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The Variance Theorem of Rogers and
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The Selection Theorem of Mahler [68
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26 The Torus Group E d /L 395 where
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Proposition 26.2. E d /L is a compa
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26 The Torus Group E d /L 399 theor
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26 The Torus Group E d /L 401 The m
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The formula to calculate the Jordan
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27 Special Problems in the Geometry
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Asymptotic Estimates 27 Special Pro
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27 Special Problems in the Geometry
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28 Basis Reduction and Polynomial A
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(ii) �b1� ≤2 1 2 (d−1) min
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28 Basis Reduction and Polynomial A
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and so 28 Basis Reduction and Polyn
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Shortest Lattice Vector Problem 28
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and therefore �x − m� 2 = �
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29 Packing of Balls and Positive Qu
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29.3 Error Correcting Codes and Bal
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30 Packing of Convex Bodies 439 The
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30 Packing of Convex Bodies 441 Nex
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Packing and Central Symmetrization
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Existence of Densest Lattice Packin
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30 Packing of Convex Bodies 447 Rem
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A Lower Estimate for the Maximum La
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Lattice Packing and Packing of Tran
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a f b o e 2D c Fig. 30.3. Disc and
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31 Covering with Convex Bodies 455
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32 Tiling with Convex Polytopes 463
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Define 32 Tiling with Convex Polyto
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33.1 Fejes Tóth’s Inequality for
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Noting that 2h 2 v − hhvv = 2π 2
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33 Optimum Quantization 485 The con
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(7) � (x, y) : x ∈ K, 0 ≤ y
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Next, we prove that (18) lim inf n
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Generalizations 33 Optimum Quantiza
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33 Optimum Quantization 493 set wit
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33 Optimum Quantization 495 In (4)
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33 Optimum Quantization 497 encoder
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34 Koebe’s Representation Theorem
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G 34 Koebe’s Representation Theor
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Di+1 34 Koebe’s Representation Th
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References 1. Achatz, H., Kleinschm
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References 515 36. Arias de Reyna,
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References 517 88. Beer, G., Topolo
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References 519 136. Bohr, H., Molle
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References 521 190. Carathéodory,
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References 523 237. Dantzig, G.B.,
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References 525 286. Edmonds, A.L.,
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References 527 337. Florian, A., Ex
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References 529 385. Goodman, J.E.,
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References 531 435. Gruber, P.M., C
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References 533 487. Heil, E., Über
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References 535 534. Hyuga, T., An e
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References 537 582. Khovanskiĭ (Ho
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References 539 632. Lazutkin, V.F.,
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References 541 684. Mani-Levitska,
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References 543 736. Minkowski, H.,
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References 545 783. Pach, J., Agarw
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References 547 835. Rickert, N.W.,
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References 549 891. Schmidt, E., Di
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References 551 941. Skubenko, B.F.,
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References 553 992. Teissier, B., V
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References 555 1042. Zassenhaus, H.
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558 INDEX Bravais classification of
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560 INDEX existence of elementary v
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562 INDEX Kneser-Macbeath theorem,
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564 INDEX rational lattice, 414 rat
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566 INDEX transference theorem of K
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568 AUTHOR INDEX Björner, 265 Blas
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570 AUTHOR INDEX Giles, J., 1 Giles
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572 AUTHOR INDEX Kuczma, 13, 17 Kü
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574 AUTHOR INDEX Pick, 316 Pisier,
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576 AUTHOR INDEX Tanemura, 464 Tarj
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578 List of Symbols sch, schL Schwa
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289. Manin: Gauge Field Theory and
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