Gruber P. Convex and Discrete Geometry

More documents

Recommendations

Info

8 The Brunn–Minkowski Inequality 141 χ(bd C)A(bd C) + 2 P(C)P(D) + A(bd C)χ(bd D) π ≥ 2 χ(C)A(D) + 4 P(C)P(D) + 2A(C)χ(D). π Since C, D ∈ C,wehaveχ(C) = χ(D) = 1 and A(bd C) = A(bd D) = 0. Hence 2π(A(C) + A(D)) − P(C)P(D) ≤ 0. Thus, if 2π(A(C) + A(D)) − P(C)P(D) >0, one of C, D contains a congruent copy of the other disc in its interior. This proves the theorem for convex polygons. Assume, second, that C, D are proper convex discs such that 2π � A(C) + (A(D) � − P(C)P(D) >0, A(C) �= A(D), say A(C) >A(D). Choose convex polygons Q ⊆ C, R ⊇ D such that 2π � A(Q) + A(R) � − P(Q)P(R) >0, A(Q) >A(R). By the first part of the proof a suitable congruent copy of R is then contained in int Q. Thus, a fortiori, a suitable congruent copy of D is contained in int C, concluding the proof of the theorem. ⊓⊔ Remark. It is an open question to extend Hadwiger’s containment theorem to higher dimensions in a simple way. For ideas in this direction due to Zhou and Zhang, see the references in Klain and Rota [587]. 8 The Brunn–Minkowski Inequality In the Brunn–Minkowski inequality, the volume V (C + D) of the Minkowski sum C + D ={x + y : x ∈ C, y ∈ D}, of two convex bodies C, D, is estimated in terms of the volumes of C and D. Around the classical inequality, a rich theory with numerous applications developed over the course of the twentieth century. In this section we first present different versions of the Brunn–Minkowski inequality, among them extensions to non-convex sets and integrals. Then, applications of the Brunn–Minkowski inequality to the classical isoperimetric inequality, sand piles, capillary surfaces, and Wulff’s theorem from crystallography are given. Finally, we show that general Brunn–Minkowski or isoperimetric type inequalities lead to the concentration of measure phenomenon. For references and related material, see Leichtweiss [640], Schneider [907], Ball [53] and, in particular, the comprehensive survey of Gardner [360] and the report of Barthe [77].
142 Convex Bodies 8.1 The Classical Brunn–Minkowski Inequality The Brunn–Minkowski inequality was first proved by Brunn [173,174] with a clever but rather vague proof. The equality case, in particular, was not settled satisfactorily, as was pointed out by Minkowski. Then, both Brunn [175] and Minkowski [735] provided correct proofs. At present several essentially different proofs are known, as well as a series of far-reaching extensions and applications. Dinghas [270] commented on Brunn’s proof as follows: The ingenious idea of Brunn to relate two convex sets by equal volume ratios ... saw substantial progress in the last twenty years. It did not yield only refinements of old, but produced also new results. The proof that will be given below is a variant of the proof of Brunn [173, 174], respectively of its precise version due to Kneser and Süss [600]. It is by induction and makes use of the idea of Brunn which relates the d-dimensional case to the (d − 1)-dimensional case by equal volume ratios: let u ∈ Sd−1 and H(t) = {x : u · x = t}. For0≤s ≤ 1lettC(s) and tD(s) be such that the hyperplanes H(tC(s)) and H(tD(s)) divide the volume of C, respectively, D, in the ratio s : 1−s. Clearly, C + D ⊇ � � � C ∩ H tC(s) � + D ∩ H � tD(s) �� . 0≤s≤1 Now apply induction to C ∩ H � tC(s) � + D ∩ H � tD(s) � and use Fubini’s theorem. A different proof is due to Blaschke [124]. It makes use of a property of Steiner symmetrization, namely that st(C + D) ⊇ st C + st D. This makes it possible to reduce the proof to the trivial case where C and D are balls. For Blaschke’s proof, see Sect. 9.2. The Classical Inequality Our aim is to prove the Brunn–Minkowski inequality: Theorem 8.1. Let C, D ∈ C. Then: (1) V (C + D) 1 d ≥ V (C) 1 d + V (D) 1 d , where equality holds if and only if one of C, D is a point, or C and D are improper and lie in parallel hyperplanes, or C and D are proper and positive homothetic. Proof. The following inequality will be used below: (2) � v 1 d−1 + w 1 � � d−1 V W � d−1 + ≥ (V v w 1 d + W 1 d ) d for v,w,V, W > 0, where equality holds if and only if v V d−1 d = w W d−1 . d
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: 140 Convex Bodies 7.5 Hadwiger’s
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: 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
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:
210 Convex Bodies Theorem 11.4. Let
Page 223 and 224:
212 Convex Bodies the maximum of th
Page 225 and 226:
214 Convex Bodies (20) mni ≥ 1 λ
Page 227 and 228:
216 Convex Bodies Prove analogous r
Page 229 and 230:
218 Convex Bodies Heuristic Princip
Page 231 and 232:
220 Convex Bodies λC + x νC + z
Page 233 and 234:
222 Convex Bodies To show that (K +
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 435 and 436:
Page 437 and 438:
29.3 Error Correcting Codes and Bal
Page 439 and 440:
Page 441 and 442:
Page 443 and 444:
Page 445 and 446:
Page 447 and 448:
Page 449 and 450:
30 Packing of Convex Bodies 439 The
Page 451 and 452:
30 Packing of Convex Bodies 441 Nex
Page 453 and 454:
Packing and Central Symmetrization
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 467 and 468:
Page 469 and 470:
Page 471 and 472:
Page 473 and 474:
32 Tiling with Convex Polytopes 463
Page 475 and 476:
Page 477 and 478:
Page 479 and 480:
Page 481 and 482:
Page 483 and 484:
Page 485 and 486:
Define 32 Tiling with Convex Polyto
Page 487 and 488:
Page 489 and 490:
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?