Gruber P. Convex and Discrete Geometry

More documents

Recommendations

Info

442 Geometry of Numbers This proves the right-hand inequality in (2). Next, consider the parallelotopes from our family which intersect (τ − σ)K . These cover (τ − σ)K and by (1) are all contained in τ K .Ifn is their number, it thus follows that #(L ∩ τ K ) ≥ n = nV(F) V (F) ≥ V � (τ − σ)K � V (F) = 2d (τ − σ) d , d(L) concluding the proof of the left-hand inequality in (2). ⊓⊔ Remark. Using the Möbius inversion formula from number theory, it can be shown that the density of the set of primitive points of a lattice L in E d is 1/(ζ(d)d(L)), where ζ(·) is the Riemann zeta function. In particular, this shows that the probability that a point of a lattice is primitive is 1/ζ(d). Next, the density of a discrete set T and the density of the family of translates of a given convex body by the vectors of T will be related: Proposition 30.2. Let C be a proper convex body and T a discrete set in E d . Then the upper density of the family {C + t : t ∈ T } is equal to V (C) times the upper density of T . Analogous statements hold for the lower density and the density, if the latter exists. Proof. Choose σ>0 such that C ⊆ σ K . Clearly, Then 1 (2τ) d ≤ � t∈T (C + t) ∩ τ K �= ∅⇒t ∈ (τ + σ)K, t ∈ (τ + σ)K ⇒ (C + t) ⊆ (τ + 2σ)K. V � (C + t) ∩ τ K � ≤ 1 (2τ) d #� T ∩ (τ + σ)K � V (C) 1 � 2(τ + 2σ) � d � V � (C + t) ∩ (τ + 2σ)K ��2(τ + 2σ) �d . t∈T (2τ) d Now let τ →∞to get the equalities for the upper and the lower density and the density. ⊓⊔ Corollary 30.1. Let C be a proper convex body and L a lattice in E d . Then the family {C + l : l ∈ L} of translates of C by the vectors of L has density δ(C, L) = V (C) d(L) . The family {C +l : l ∈ L} of translates of the body C by the vectors of the lattice L is sometimes called a set lattice with set C and lattice L and δ(C, L) is its density.
Packing and Central Symmetrization 30 Packing of Convex Bodies 443 Given a convex body C, its central symmetrization is the convex body D = 1 (C − C) 2 � = � 1 (x − y) : x, y ∈ C 2 �� . (The convex body 2D = C −C is called the difference body of C.) Minkowski [740] discovered that lattice packings of a convex body and its difference body are closely related. His proof yields the following, slightly more general result. Proposition 30.3. Let C be a proper convex body, D = 1 2 (C − C) its central symmetrization and T a discrete set in Ed . Then the following statements are equivalent: (i) {C + t : t ∈ T } is a packing. (ii) {D + t : t ∈ T } is a packing. Proof. It is sufficient to show the following equivalence: (3) Let s, t ∈ E d . Then We shall use the equalities (int C + s) ∩ (int C + t) �= ∅⇔(int D + s) ∩ (int D + t) �= ∅. (4) int D = 1 (int C − int C), 2 (5) int C = 1 (int C + int C). 2 Then (int C + s) ∩ (int C + t) �= ∅ ⇒ x + s = y + t for suitable x, y ∈ int C ⇒ 1 1 (x − y) + s = (y − x) + t 2 2 ⇒ (int D + s) ∩ (int D + t) �= ∅by (4) ⇒ 1 1 (u − v) + s = (w − z) + t for suitable u,v,w,z ∈ int C by (4) 2 2 ⇒ 1 1 (u + z) + s = (w + v) + t 2 2 ⇒ (int C + s) ∩ (int C + t) �= ∅by (5), concluding the proof of (3) and thus of the proposition. ⊓⊔ Remark. If L is a packing lattice of C or, equivalently, of D = 1 2 (C − C), the corresponding densities are δ(C, L) = V (C) d(L) and δ(D, L) = V (D) d(L) ,
Page 1 and 2:
Grundlehren der mathematischen Wiss
Page 3 and 4:
Peter M. Gruber Institute of Discre
Page 5 and 6:
VI Preface theory of finite groups
Page 7 and 8:
Contents Preface ..................
Page 9 and 10:
Contents XI 12 Special Convex Bodie
Page 11 and 12:
Contents XIII 29 Packing of Balls a
Page 13 and 14:
2 Convex Functions 1 Convex Functio
Page 15 and 16:
4 Convex Functions x epi f f C y Fi
Page 17 and 18:
6 Convex Functions Together these i
Page 19 and 20:
8 Convex Functions Theorem 1.4. Let
Page 21 and 22:
10 Convex Functions z = f (x) + f
Page 23 and 24:
12 Convex Functions 1.3 Convexity C
Page 25 and 26:
14 Convex Functions Proof (by affin
Page 27 and 28:
16 Convex Functions Minkowski’s I
Page 29 and 30:
18 Convex Functions (ii) Let x > 0.
Page 31 and 32:
20 Convex Functions 2 Convex Functi
Page 33 and 34:
22 Convex Functions | f (z) − f (
Page 35 and 36:
24 Convex Functions First-Order Dif
Page 37 and 38:
26 Convex Functions Remark. More pr
Page 39 and 40:
28 Convex Functions even more is tr
Page 41 and 42:
30 Convex Functions G(x). The proof
Page 43 and 44:
32 Convex Functions by (15), unifor
Page 45 and 46:
34 Convex Functions 2.4 A Stone-Wei
Page 47 and 48:
36 Convex Functions x x Fig. 2.1. S
Page 49 and 50:
38 Convex Functions F is convex. Us
Page 51 and 52:
40 Convex Bodies results, character
Page 53 and 54:
42 Convex Bodies Convex Hulls Given
Page 55 and 56:
44 Convex Bodies The next result is
Page 57 and 58:
46 Convex Bodies C ∩ L ⊥ clearl
Page 59 and 60:
48 Convex Bodies Suppose not. Then,
Page 61 and 62:
50 Convex Bodies The Centrepoint of
Page 63 and 64:
52 Convex Bodies Since G is open, w
Page 65 and 66:
54 Convex Bodies in one of the two
Page 67 and 68:
56 Convex Bodies or geometric infor
Page 69 and 70:
58 Convex Bodies v HC (u, h(u)) (v,
Page 71 and 72:
60 Convex Bodies C D Fig. 4.3. Sepa
Page 73 and 74:
62 Convex Bodies For the proof of t
Page 75 and 76:
64 Convex Bodies in contradiction t
Page 77 and 78:
66 Convex Bodies (6) R(t) ⊆ E d i
Page 79 and 80:
68 Convex Bodies 5 The Boundary of
Page 81 and 82:
70 Convex Bodies for all y ∈ C an
Page 83 and 84:
72 Convex Bodies the existence of s
Page 85 and 86:
74 Convex Bodies where r(·) is a s
Page 87 and 88:
76 Convex Bodies Hence x ∈ conv e
Page 89 and 90:
78 Convex Bodies N = � � X, O Y
Page 91 and 92:
80 Convex Bodies Further topics of
Page 93 and 94:
82 Convex Bodies We are now ready t
Page 95 and 96:
84 Convex Bodies similarly, D j is
Page 97 and 98:
86 Convex Bodies For the proof of t
Page 99 and 100:
88 Convex Bodies The definition of
Page 101 and 102:
90 Convex Bodies If ei ∈ Pi is no
Page 103 and 104:
92 Convex Bodies Let v(·) denote (
Page 105 and 106:
94 Convex Bodies The present sectio
Page 107 and 108:
96 Convex Bodies Proposition 6.5. L
Page 109 and 110:
98 Convex Bodies Thus Second, Minko
Page 111 and 112:
100 Convex Bodies The Valuation Pro
Page 113 and 114:
102 Convex Bodies Minkowski’s the
Page 115 and 116:
104 Convex Bodies To remedy the sit
Page 117 and 118:
106 Convex Bodies Proof. Applying S
Page 119 and 120:
108 Convex Bodies Third, let C be a
Page 121 and 122:
110 Convex Bodies A Problem of Sant
Page 123 and 124:
112 Convex Bodies sets of S. Ifφ c
Page 125 and 126:
114 Convex Bodies = � (−1) |J|
Page 127 and 128:
116 Convex Bodies If P has dimensio
Page 129 and 130:
118 Convex Bodies Since, by Volland
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 and 180:
168 Convex Bodies Clearly, the inte
Page 181 and 182:
170 Convex Bodies Note that �st :
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 197 and 198:
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 221 and 222:
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 +
Page 235 and 236:
224 Convex Bodies The integral here
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 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 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
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
Page 347 and 348:
336 Convex Polytopes economics, von
Page 349 and 350:
338 Convex Polytopes Existence of S
Page 351 and 352:
340 Convex Polytopes Description of
Page 353 and 354:
342 Convex Polytopes (6) u piai xq
Page 355 and 356:
344 Convex Polytopes For x ∈ Bd
Page 357 and 358:
346 Convex Polytopes Rational Polyh
Page 359 and 360:
348 Convex Polytopes Totally Dual I
Page 361 and 362:
350 Convex Polytopes of the vectors
Page 363 and 364:
Geometry of Numbers and Aspects of
Page 365 and 366:
Extension to Crystallographic Group
Page 367 and 368:
Relations Between Different Bases 2
Page 369 and 370:
21 Lattices 359 triangle conv{o,(u,
Page 371 and 372:
By definition of bi+1, Thus (6) yie
Page 373 and 374:
21 Lattices 363 A similar, slightly
Page 375 and 376:
21.4 Polar Lattices 21 Lattices 365
Page 377 and 378:
The First Fundamental Theorem 22 Mi
Page 379 and 380:
22 Minkowski’s First Fundamental
Page 381 and 382:
If 22 Minkowski’s First Fundament
Page 383 and 384:
22 Minkowski’s First Fundamental
Page 385 and 386:
Proof. Consider the linear forms l1
Page 387 and 388:
Proof. It is sufficient to consider
Page 389 and 390:
Thus V (C) ≥ V (O) = 2d � �
Page 391 and 392:
Given C and L, the problem arises t
Page 393 and 394:
23 Successive Minima 383 as require
Page 395 and 396:
24 The Minkowski-Hlawka Theorem 24
Page 397 and 398:
(6) (V (J) ≈) � n�= 0 (7) J
Page 399 and 400:
24 The Minkowski-Hlawka Theorem 389
Page 401 and 402: The Variance Theorem of Rogers and
Page 403 and 404: The Selection Theorem of Mahler [68
Page 405 and 406: 26 The Torus Group E d /L 395 where
Page 407 and 408: Proposition 26.2. E d /L is a compa
Page 409 and 410: 26 The Torus Group E d /L 399 theor
Page 411 and 412: 26 The Torus Group E d /L 401 The m
Page 413 and 414: The formula to calculate the Jordan
Page 415 and 416: 27 Special Problems in the Geometry
Page 417 and 418: Asymptotic Estimates 27 Special Pro
Page 419 and 420: 27 Special Problems in the Geometry
Page 421 and 422: 28 Basis Reduction and Polynomial A
Page 423 and 424: (ii) �b1� ≤2 1 2 (d−1) min
Page 425 and 426: 28 Basis Reduction and Polynomial A
Page 427 and 428: and so 28 Basis Reduction and Polyn
Page 429 and 430: Shortest Lattice Vector Problem 28
Page 431 and 432: and therefore �x − m� 2 = �
Page 433 and 434: 29 Packing of Balls and Positive Qu
Page 437 and 438: 29.3 Error Correcting Codes and Bal
Page 449 and 450: 30 Packing of Convex Bodies 439 The
Page 451: 30 Packing of Convex Bodies 441 Nex
Page 455 and 456: Existence of Densest Lattice Packin
Page 457 and 458: 30 Packing of Convex Bodies 447 Rem
Page 459 and 460: A Lower Estimate for the Maximum La
Page 461 and 462: Lattice Packing and Packing of Tran
Page 463 and 464: a f b o e 2D c Fig. 30.3. Disc and
Page 465 and 466: 31 Covering with Convex Bodies 455
Page 473 and 474: 32 Tiling with Convex Polytopes 463
Page 485 and 486: Define 32 Tiling with Convex Polyto
Page 491 and 492: 33.1 Fejes Tóth’s Inequality for
Page 493 and 494: Noting that 2h 2 v − hhvv = 2π 2
Page 495 and 496: 33 Optimum Quantization 485 The con
Page 497 and 498: (7) � (x, y) : x ∈ K, 0 ≤ y
Page 499 and 500: Next, we prove that (18) lim inf n
Page 501 and 502: Generalizations 33 Optimum Quantiza
Page 503 and 504:
33 Optimum Quantization 493 set wit
Page 505 and 506:
33 Optimum Quantization 495 In (4)
Page 507 and 508:
33 Optimum Quantization 497 encoder
Page 509 and 510:
34 Koebe’s Representation Theorem
Page 511 and 512:
G 34 Koebe’s Representation Theor
Page 513 and 514:
Di+1 34 Koebe’s Representation Th
Page 515 and 516:
Page 517 and 518:
Page 519 and 520:
Page 521 and 522:
Page 523 and 524:
References 1. Achatz, H., Kleinschm
Page 525 and 526:
References 515 36. Arias de Reyna,
Page 527 and 528:
References 517 88. Beer, G., Topolo
Page 529 and 530:
References 519 136. Bohr, H., Molle
Page 531 and 532:
References 521 190. Carathéodory,
Page 533 and 534:
References 523 237. Dantzig, G.B.,
Page 535 and 536:
References 525 286. Edmonds, A.L.,
Page 537 and 538:
References 527 337. Florian, A., Ex
Page 539 and 540:
References 529 385. Goodman, J.E.,
Page 541 and 542:
References 531 435. Gruber, P.M., C
Page 543 and 544:
References 533 487. Heil, E., Über
Page 545 and 546:
References 535 534. Hyuga, T., An e
Page 547 and 548:
References 537 582. Khovanskiĭ (Ho
Page 549 and 550:
References 539 632. Lazutkin, V.F.,
Page 551 and 552:
References 541 684. Mani-Levitska,
Page 553 and 554:
References 543 736. Minkowski, H.,
Page 555 and 556:
References 545 783. Pach, J., Agarw
Page 557 and 558:
References 547 835. Rickert, N.W.,
Page 559 and 560:
References 549 891. Schmidt, E., Di
Page 561 and 562:
References 551 941. Skubenko, B.F.,
Page 563 and 564:
References 553 992. Teissier, B., V
Page 565 and 566:
References 555 1042. Zassenhaus, H.
Page 567 and 568:
558 INDEX Bravais classification of
Page 569 and 570:
560 INDEX existence of elementary v
Page 571 and 572:
562 INDEX Kneser-Macbeath theorem,
Page 573 and 574:
564 INDEX rational lattice, 414 rat
Page 575 and 576:
566 INDEX transference theorem of K
Page 577 and 578:
568 AUTHOR INDEX Björner, 265 Blas
Page 579 and 580:
570 AUTHOR INDEX Giles, J., 1 Giles
Page 581 and 582:
572 AUTHOR INDEX Kuczma, 13, 17 Kü
Page 583 and 584:
574 AUTHOR INDEX Pick, 316 Pisier,
Page 585 and 586:
576 AUTHOR INDEX Tanemura, 464 Tarj
Page 587 and 588:
578 List of Symbols sch, schL Schwa
Page 589:
289. Manin: Gauge Field Theory and
show all

Gruber P. Convex and Discrete Geometry

Create successful ePaper yourself

Delete template?

Save as template?