Handbook of convex geometry.

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 6
Contens of Volume A......Page 8
Contents of Volume B......Page 9
List of Contributors......Page 12
Part 3, Discrete Aspects of Convexity......Page 14
3.1. Geometry of numbers......Page 16
2. Lattices and the space of lattices......Page 18
3. The fundamental theorems......Page 21
4. The Minkowski-Hlawka theorem......Page 25
5. Successive minima......Page 27
6. Reduction theory......Page 28
6.1. Korkin-Zolotarev reduction......Page 29
6.2. LLL or Lovdsz reduction......Page 30
7.1. Lattice constants......Page 31
7.3. Moment problem......Page 32
7.5. Conjecture on the product of non-homogeneous linear forms......Page 33
References......Page 34
3.2. Lattice points......Page 42
1.1. Introduction......Page 44
1.2. General definitions and notation......Page 45
2.1. Minkowski's fundamental theorem......Page 46
2.2. Successive minima......Page 47
3.1. Lattice points and intrinsic volumes......Page 48
3.2. Classes of lattices......Page 51
3.3. Nonlinear inequalities......Page 53
4.1. General lattice polytopes......Page 55
4.2. Convex lattice polytopes: Equalities......Page 56
4.3. Convex lattice polytopes: Inequalities......Page 58
5. Lattice polyhedra in combinatorial optimization......Page 59
5.1. The combinatorics of associated lattice polyhedra......Page 60
5.2. Polyhedral combinatorics......Page 61
6.1. Algorithmic problems in geometry of numbers......Page 63
6.2. NP-hard problems......Page 64
6.3. Polynomial time solvability......Page 65
6.4. The complexity of computing upper and lower bound functionals......Page 66
References......Page 67
3.3. Packing and covering with convex sets......Page 76
1. Introduction......Page 78
2.1. Densities......Page 80
2.2. Existence of efficient arrangements......Page 83
2.3. Upper bounds for S(Bd) and lower bounds for j4(Bd)......Page 87
2.4. Constructive methods for dense packings......Page 91
2.5. Lattice arrangements......Page 95
2.6. Multiple arrangements......Page 98
3.1. Spherical space......Page 101
3.2. Hyperbolic space......Page 108
4.1. Existence of efficient arrangements......Page 111
4.2. Upper bounds for packing densities and lower bounds for covering densities......Page 114
4.3. Additional results on density bounds......Page 117
5.1. Other concepts of efficiency of arrangements......Page 120
5.2. Density bounds for arrangements with restricting conditions......Page 122
5.3. Newton number and related problems......Page 126
References......Page 127
3.4. Finite packing and covering......Page 138
1.2. General packing and covering problems......Page 140
1.3. The scope of this survey......Page 141
2. Sausage problems......Page 142
2.1. Densities......Page 143
2.2. Sphere packings and sphere coverings......Page 144
2.3. General convex bodies......Page 148
2.4. Algorithmic aspects......Page 149
2.5.1. Lattice covering......Page 153
2.5.3. Dispersion......Page 154
2.5.4. Crystallography......Page 155
3.1. Bin packing problems and their relatives......Page 156
3.2. Approximation algorithms......Page 159
3.3. Some applications......Page 161
4.1. Some container problems......Page 162
4.2.2. Minimum energy packing......Page 164
4.2.4. Insphere, circumsphere and related problems......Page 165
4.3. The Koebe-Andreev-Thurston theorem and its relatives......Page 166
4.4. Packing and covering in combinatonal optimization......Page 167
References......Page 168
3.5. Tilings......Page 176
1. Introduction......Page 178
2. Basic notions......Page 179
3.1. Tilings by regular polygons......Page 183
3.2. Some general results on well-behaved plane tilings......Page 185
3.3. Classification with respect to topological transitivity properties......Page 189
3.4. Classification with respect to symmetries......Page 190
4. Monohedral tilings......Page 192
4.1. Lattice tilings......Page 193
4.2. Prototiles of monohedral tilings......Page 195
4.3. Monotypic tilings......Page 197
5. Non-periodictilings......Page 198
References......Page 203
3.6. Valuations and dissections......Page 210
1.1. Euclidean notions......Page 212
1.2. Valuations......Page 213
1.3. Extensions......Page 215
2.1. Volume and derived valuations......Page 216
2.3. The lattice point enumerator......Page 220
3.1. The algebra structure......Page 221
3.2. Negative dilatations and Eulcr-type relations......Page 226
3.4. The cone group......Page 227
3.5. Translation covariance......Page 228
3.6. Mixed polytopes......Page 230
3.8. Invariance under other groups......Page 231
4.1. The algebra of polytopcs......Page 232
4.2. Cones and angles......Page 234
4.3. The pol'tope groups......Page 236
4.4. Spherical dissections......Page 237
4.5. Hilbcrt's third problem......Page 242
5. Characterization theorems......Page 245
5.1. Continuity and monotonicity......Page 246
5.2. Minkowski additive functions......Page 247
5.3. Volume and moment......Page 250
5.4. Intrinsic volumes and moments......Page 252
5.5. Translation invariance and covariance......Page 253
5.6. Lattice invariant valuations......Page 256
References......Page 257
3.7. Geometric crystallography......Page 266
1.1. Basic definitions......Page 268
1.2. Isometries......Page 269
1.3. Groups of symmetry operations......Page 270
1.4. Regularity condition for an (r, R)-system......Page 272
2.1. Definition and basic properties of Dirichlet domains......Page 273
2.2. Characterization of Dirichlet domains......Page 275
2.3. Dirichlet domain partition......Page 276
2.4. Regularity condition for normal space partitions......Page 278
3. Translation lattices......Page 279
3.1. Lattice bases......Page 280
3.2. Lattice planes......Page 281
3.3. Parallelotopes......Page 282
3.4. Results on parallelotopes......Page 284
3.5. Determination of a lattice basis......Page 285
4.1. Definition of the Z-reduced form......Page 286
4.2. The reduction scheme of Lagrange......Page 287
4.3. The reduction scheme of Seeber......Page 288
4.4. The reduction scheme of Selling......Page 289
4.5. The reduction scheme of Minkowski......Page 290
4.6. Optimal bases......Page 291
5.1. Crystallographic symmetry operations......Page 292
5.2. Crystallographic point groups......Page 295
5.3. Symmetry of translation lattices......Page 297
5.4. Crystal forms......Page 299
6. Infinite groups of symmetry operations......Page 300
6.1. Space groups......Page 301
6.2. Crystallographic orbits......Page 303
6.3. Packing of balls......Page 304
6.4. Colour groups......Page 306
6.5. Subpenodic groups......Page 307
7.1. Definitions......Page 308
7.2. The projection method......Page 309
References......Page 310
Part 4, Analytic Aspects of Convexity......Page 320
4.1. Convexity and differential geometry......Page 322
Introduction......Page 324
1. Differential geometric characterization of convexity......Page 325
2. Elementary symmetric functions of principal curvatures respectively principal radii of curvature at Euler points......Page 330
3. Mixed discriminants and mixed volumes......Page 336
4. Differential geometric proof of the Aleksandrov-Fenchel-Jessen inequalities......Page 342
5. Uniqueness theorems for convex hypersurfaces......Page 347
6. Convexity and relative geometry......Page 352
7. Convexity and affine differential geometry......Page 354
References......Page 355
4.2. Convex functions......Page 358
1.1. Mid-convexity and continuity......Page 361
1.2. Lower semi-continuity and closure of convex functions......Page 362
1.3. Conjugate convex functions......Page 365
2.1. Functions defined on R......Page 367
2.2. Related concepts......Page 369
2.3. Functions defined on a linear space I......Page 372
2.4. Differentiable convex functions......Page 376
3.1. Classical inequalities......Page 377
3.2. Matrix inequalities......Page 378
References......Page 380
4.3. Convexity and calculus of variations......Page 382
Introduction......Page 384
1.2. The second derivative test......Page 385
2.1. Examples......Page 386
2.2. A necessary condition: The Euler-Lagrange equation......Page 387
2.3. Convex integrals......Page 388
2.4. Further sufficient conditions......Page 390
2.5. Broken extremals......Page 392
2.6. Existence and regularity. Lower semicontinuity......Page 393
2.7. Relaxation and convexification......Page 396
2.8. Problems in parametric form. Geometric theory......Page 397
3. Multiple integrals in the calculus of variations......Page 399
3.1. Convex integrals......Page 400
3.2. The necessary conditions of Weierstrass and Legendre......Page 401
3.3. Sufficient conditions with invariant integrals......Page 402
3.4. Lower semicontinuity and existence......Page 403
3.5. Problems in parametric form......Page 404
References......Page 406
4.4. On isoperimetric theorems of mathematical physics......Page 408
2. Historical and bibliographical comments......Page 410
3. Rearrangements......Page 411
4. Capacity......Page 414
5. Torsional rigidity......Page 416
6. Clamped membranes......Page 418
7. Clamped plates......Page 420
References......Page 422
4.5. The local theory of normed spaces and its applications to convexity......Page 426
1. Introduction......Page 428
2.1. A few classic facts and definitions......Page 429
2.2. Type, cotype and related notions......Page 432
2.3. Ideal norms and p-absolutely summing operators......Page 439
3.1. Hilbertian subspaces......Page 442
3.2. Embedding of and into tp......Page 448
4.1. John's theorem and ellipsoids related to it......Page 452
4.2. Inertia ellipsoids, isotropic position and generalized Khintchine inequality......Page 455
4.3. M ellipsoids and the inverse Brunn-Minkowski inequality......Page 459
5.1. Distances from and projections on classical spaces, especially t2......Page 462
5.2. Random normed spaces and existence of bases......Page 468
5.3. Random operators and the distances between spaces of some special families......Page 472
6.1. Isomorphic symmetrization and its applications......Page 477
6.2. Zonoids and Minkowski sums, approximation and symmetrization......Page 481
6.3. Some additional results......Page 486
References......Page 488
4.6. Nonexpansive maps and fixed points......Page 498
1.1. Main definitions......Page 500
2. Some examples......Page 501
3. Some results (and some history)......Page 502
3.1. Main results......Page 503
3.3. Asymptotic centers and minimal invariant sets......Page 504
4.1. Generalizing the class of mappings......Page 505
4.2. Generalizing the space......Page 506
4.5. Fixed points and unbounded sets......Page 507
5.2. The structure of the fixed point set......Page 508
6. Some other general facts......Page 509
6.2. Some general references......Page 510
References......Page 511
4.7. Critical exponents......Page 514
1. Motivation......Page 516
2. History......Page 520
3. Hilbert space......Page 521
4. Polytopes......Page 525
5. Open problems......Page 530
References......Page 532
4.8. Fourier series and spherical harmonics in convexity......Page 536
1.1. Convex bodies......Page 538
1.2. The Laplace-Beltrami operator......Page 540
1.3. Fourier series......Page 541
1.4. Spherical harmonics......Page 542
2.1. The work of Hurwitz on the isoperimetric inequality......Page 545
2.2. The Fourier expansion of the support function and mixed area inequalities......Page 546
2.3. Circumscribed polygons and rotors......Page 548
2.4. Other geometric applications of Fourier series......Page 552
3.1. The harmonic expansion of the support function......Page 554
3.3. More about functions on the sphere with vanishing integrals over great circles......Page 555
3.4. Projections of convex bodies and related matters......Page 557
3.5. Functions on the sphere with vanishing integrals over hemispheres......Page 560
3.6. Rotors in polytopes......Page 561
3.7. Inequalities for mean projection measures and mixed volumes......Page 562
3.8. Wirtinger's inequality and its applications......Page 564
3.9. Other geometric applications of spherical harmonics......Page 566
References......Page 567
4.9. Zonoids and generalisations......Page 574
2. Basic definitions and properties......Page 576
3. Analytic characterisations of zonoids......Page 582
4. Centrally symmetric bodies and the spherical Radon transform......Page 584
5. Projections onto hyperplanes......Page 589
6. Projection functions on higher rank Grassmannians......Page 592
7. Classes of centrally symmetric bodies......Page 594
8. Zonoids in integral and stochastic geometry......Page 596
References......Page 598
4.10. Baire categories in convexity......Page 604
2. A typical proof of a Baire category type result in convexity......Page 606
3. Boundary properties of arbitrary convex bodies......Page 607
4. Smoothness and strict convexity......Page 608
5. Geodesics......Page 610
7. Normals, mirrors and diameters......Page 611
8. Approximation of convex bodies by polytopes......Page 612
10. Shadow boundaries......Page 614
11. Metric projections......Page 615
12. Miscellaneous results for typical convex bodies......Page 616
13. Starbodies, starsets and compact sets......Page 617
References......Page 618
Part 5, Stochastic Aspects of Convexity......Page 624
5.1. Integral geometry......Page 626
1. Preliminaries: Spaces, groups, and measures......Page 628
2. Intersection formulae......Page 630
3. Minkowski addition and projections......Page 637
4. Distance integrals and contact measures......Page 642
5. Extension to the convex ring......Page 648
6. Translative integral geometry and auxiliary zonoids......Page 652
7. Lines and flats through convex bodies......Page 656
References......Page 663
5.2. Stochastic geometry......Page 668
Preliminaries.......Page 670
1. Random points in a convex body......Page 672
2. Random flats intersecting a convex body......Page 679
3. Randoin convex bodies.......Page 681
4. Random sets......Page 684
5. Point processes.......Page 687
6. Random surfaces......Page 696
7. Random mosaics.......Page 700
8. Stereology......Page 705
References.......Page 708
Author Index ......Page 716
Subject Index ......Page 750
Back Cover......Page 770