Classical topics in discrete geometry:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY [u.a.]
Springer
2010
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| Schriftenreihe: | CMS Books in Mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. [153] - 163 |
| Beschreibung: | XIII, 163 S. |
| ISBN: | 9781441905994 |
Internformat
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| 245 | 1 | 0 | |a Classical topics in discrete geometry |c Károly Bezdek |
| 264 | 1 | |a New York, NY [u.a.] |b Springer |c 2010 | |
| 300 | |a XIII, 163 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a CMS Books in Mathematics | |
| 500 | |a Literaturverz. S. [153] - 163 | ||
| 650 | 4 | |a Discrete geometry | |
| 650 | 0 | 7 | |a Diskrete Geometrie |0 (DE-588)4130271-0 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-020537199 | ||
Datensatz im Suchindex
| _version_ | 1804143225182617601 |
|---|---|
| adam_text | Titel: Classical topics in discrete geometry
Autor: Bezdek, Károly
Jahr: 2010
Contents
Preface........................................................VII
Part I Classical Topics Revisited
1 Sphere Packings........................................... 3
1.1 Kissing Numbers of Spheres .............................. 3
1.2 One-Sided Kissing Numbers of Spheres..................... 5
1.3 On the Contact Numbers of Finite Sphere Packings.......... 6
1.4 Lower Bounds for the (Surface) Volume of Voronoi Cells in
Sphere Packings......................................... 7
1.5 On the Density of Sphere Packings in Spherical Containers ... 12
1.6 Upper Bounds on Sphere Packings in High Dimensions....... 13
1.7 Uniform Stability of Sphere Packings....................... 15
2 Finite Packings by Translates of Convex Bodies ........... 17
2.1 Hadwiger Numbers of Convex Bodies...................... 17
2.2 One-Sided Hadwiger Numbers of Convex Bodies............. 18
2.3 Touching Numbers of Convex Bodies....................... 19
2.4 On the Number of Touching Pairs in Finite Packings......... 20
3 Coverings by Homothetic Bodies - Illumination and
Related Topics............................................. 23
3.1 The Illumination Conjecture.............................. 23
3.2 Equivalent Formulations.................................. 24
3.3 The Illumination Conjecture in Dimension Three............ 24
3.4 The Illumination Conjecture in High Dimensions............ 25
3.5 On the X-Ray Number of Convex Bodies................... 28
3.6 The Successive Illumination Numbers of Convex Bodies...... 29
3.7 The Illumination and Covering Parameters of Convex Bodies . 31
3.8 On the Vertex Index of Convex Bodies..................... 32
4 Coverings by Planks and Cylinders........................ 35
4.1 Plank Theorems......................................... 35
4.2 Covering Convex Bodies by Cylinders...................... 37
4.3 Covering Lattice Points by Hyperplanes.................... 39
4.4 On Some Strengthenings of the Plank Theorems of Ball and
Bang .................................................. 41
4.5 On Partial Coverings by Planks: Bang s Theorem Revisited ... 43
5 On the Volume of Finite Arrangements of Spheres......... 47
5.1 The Conjecture of Kneser and Poulsen..................... 47
5.2 The Kneser-Poulsen Conjecture for Continuous Contractions . 48
5.3 The Kneser-Poulsen Conjecture in the Plane................ 49
5.4 Non-Euclidean Kneser-Poulsen-Type Results ............... 51
5.5 Alexander s Conjecture................................... 53
5.6 Densest Finite Sphere Packings ........................... 54
6 Ball-Polyhedra as Intersections of Congruent Balls ........ 57
6.1 Disk-Polygons and Ball-Polyhedra......................... 57
6.2 Shortest Billiard Trajectories in Disk-Polygons.............. 57
6.3 Blaschke-Lebesgue-Type Theorems for Disk-Polygons........ 59
6.4 On the Steinitz Problem for Ball-Polyhedra................. 61
6.5 On Global Rigidity of Ball-Polyhedra...................... 62
6.6 Separation and Support for Spindle Convex Sets............. 63
6.7 Caratheodory- and Steinitz-Type Results................... 65
6.8 Illumination of Ball-Polyhedra............................ 65
6.9 The Euler-Poincare Formula for Ball-Polyhedra............. 67
Part II Selected Proofs
7 Selected Proofs on Sphere Packings........................ 71
7.1 Proof of Theorem 1.3.5................................... 71
7.1.1 A proof by estimating the surface area of unions of balls 71
7.1.2 On the densest packing of congruent spherical caps of
special radius..................................... 73
7.2 Proof of Theorem 1.4.7................................... 73
7.2.1 The Voronoi star of a Voronoi cell in unit ball packings 73
7.2.2 Estimating the volume of a Voronoi star from below ... 74
7.3 Proof of Theorem 1.4.8................................... 75
7.3.1 Basic metric properties of Voronoi cells in unit ball
packings ......................................... 75
7.3.2 Wedges of types I, II, and III, and truncated wedges
of types I, and II.................................. 76
7.3.3 The lemma of comparison and a characterization of
regular polytopes.................................. 79
7.3.4 Volume formulas for (truncated) wedges.............. 80
7.3.5 The integral representation of surface density in
(truncated) wedges................................ 81
7.3.6 Truncation of wedges increases the surface density..... 84
7.3.7 Maximum surface density in truncated wedges of type I 85
7.3.8 An upper bound for the surface density in truncated
wedges of type II.................................. 86
7.3.9 The overall estimate of surface density in Voronoi cells . 88
7.4 Proof of Theorem 1.7.3................................... 89
7.4.1 The signed volume of convex polytopes............... 89
7.4.2 The volume force of convex polytopes................ 90
7.4.3 Critical volume condition........................... 91
7.4.4 Strictly locally volume expanding convex polytopes .... 92
7.4.5 From critical volume condition and infinitesimal
rigidity to uniform stability of sphere packings........ 94
8 Selected Proofs on Finite Packings of Translates of
Convex Bodies............................................. 95
8.1 Proof of Theorem 2.2.1................................... 95
8.1.1 Monotonicity of a special integral function............ 95
8.1.2 A proof by slicing via the Brunn-Minkowski inequality . 96
8.2 Proof of Theorem 2.4.3................................... 98
9 Selected Proofs on Illumination and Related Topics........101
9.1 Proof of Corollary 3.4.2 Using Rogers Classical Theorem on
Economical Coverings....................................101
9.2 Proof of Theorem 3.5.2 via the Gauss Map .................102
9.3 Proof of Theorem 3.5.3 Using Antipodal Spherical Codes of
Small Covering Radii....................................103
9.4 Proofs of Theorem 3.8.1 and Theorem 3.8.3.................106
9.4.1 From the Banach-Mazur distance to the vertex index .. 106
9.4.2 Calculating the vertex index of Euclidean balls in
dimensions 2 and 3................................107
9.4.3 A lower bound for the vertex index using the
Blaschke-Santalo inequality and an inequality of Ball
and Pajor........................................112
9.4.4 An upper bound for the vertex index using a theorem
of Rudelson.......................................H3
10 Selected Proofs on Coverings by Planks and Cylinders.....115
10.1 Proof of Theorem 4.1.7...................................115
10.1.1 On coverings of convex bodies by two planks..........115
10.1.2 A proof of the affine plank conjecture of Bang for
non-overlapping cuts...............................116
10.2 Proof of Theorem 4.2.2...................................117
10.2.1 Covering ellipsoids by 1-codimensional cylinders.......117
10.2.2 Covering convex bodies by cylinders of given
codimension......................................118
10.3 Proof of Theorem 4.5.2...................................119
10.4 Proof of Theorem 4.5.8...................................119
11 Selected Proofs on the Kneser-Poulsen Conjecture........121
11.1 Proof of Theorem 5.3.2 on the Monotonicity of Weighted
Surface Volume .........................................121
11.2 Proof of Theorem 5.3.3 on Weighted Surface and Codimension
Two Volumes...........................................124
11.3 Proof of Theorem 5.3.4 - the Leapfrog Lemma ..............126
11.4 Proof of Theorem 5.4.1...................................127
11.4.1 The spherical leapfrog lemma.......................127
11.4.2 Smooth contractions via Schlafli s differential formula .. 127
11.4.3 Relating higher-dimensional spherical volumes to
lower-dimensional ones.............................128
11.4.4 Putting pieces together.............................129
11.5 Proof of Theorem 5.4.6...................................130
11.5.1 Monotonicity of the volume of hyperbolic simplices .... 130
11.5.2 From Andreev s theorem to smooth one-parameter
family of hyperbolic polyhedra......................133
12 Selected Proofs on Ball-Polyhedra.........................135
12.1 Proof of Theorem 6.2.1...................................135
12.1.1 Finite sets that cannot be translated into the interior
of a convex body..................................135
12.1.2 From generalized billiard trajectories to shortest ones .. 137
12.2 Proofs of Theorems 6.6.1, 6.6.3, and 6.6.4...................138
12.2.1 Strict separation by spheres of radii at most one.......138
12.2.2 Characterizing spindle convex sets...................139
12.2.3 Separating spindle convex sets......................139
12.3 Proof of Theorem 6.7.1...................................140
12.3.1 On the boundary of spindle convex hulls in terms of
supporting spheres ................................140
12.3.2 From the spherical Caratheodory theorem to an
analogue for spindle convex hulls....................141
12.4 Proof of Theorem 6.8.3...................................142
12.4.1 On the boundary of spindle convex hulls in terms of
normal images....................................142
12.4.2 On the Euclidean diameter of spindle convex hulls and
normal images....................................143
12.4.3 An upper bound for the illumination number based on
a probabilistic approach............................144
12.4.4 Schramm s lower bound for the proper measure of
polars of sets of given diameter in spherical space......145
12.4.5 An upper bound for the number of sets of given
diameter that are needed to cover spherical space......147
12.4.6 The final upper bound for the illumination number .... 148
12.5 Proof of Theorem 6.9.1...................................148
12.5.1 The CW-decomposition of the boundary of a standard
ball-polyhedron...................................148
12.5.2 On the number of generating balls of a standard
ball-polyhedron...................................149
12.5.3 Basic properties of face lattices of standard
ball-polyhedra....................................150
References.....................................................153
|
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| author | Bezdek, Károly 1955- |
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| id | DE-604.BV036617090 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:44:16Z |
| institution | BVB |
| isbn | 9781441905994 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020537199 |
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| physical | XIII, 163 S. |
| publishDate | 2010 |
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| publisher | Springer |
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| spelling | Bezdek, Károly 1955- Verfasser (DE-588)1163838543 aut Classical topics in discrete geometry Károly Bezdek New York, NY [u.a.] Springer 2010 XIII, 163 S. txt rdacontent n rdamedia nc rdacarrier CMS Books in Mathematics Literaturverz. S. [153] - 163 Discrete geometry Diskrete Geometrie (DE-588)4130271-0 gnd rswk-swf Diskrete Geometrie (DE-588)4130271-0 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4419-0600-7 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020537199&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bezdek, Károly 1955- Classical topics in discrete geometry Discrete geometry Diskrete Geometrie (DE-588)4130271-0 gnd |
| subject_GND | (DE-588)4130271-0 |
| title | Classical topics in discrete geometry |
| title_auth | Classical topics in discrete geometry |
| title_exact_search | Classical topics in discrete geometry |
| title_full | Classical topics in discrete geometry Károly Bezdek |
| title_fullStr | Classical topics in discrete geometry Károly Bezdek |
| title_full_unstemmed | Classical topics in discrete geometry Károly Bezdek |
| title_short | Classical topics in discrete geometry |
| title_sort | classical topics in discrete geometry |
| topic | Discrete geometry Diskrete Geometrie (DE-588)4130271-0 gnd |
| topic_facet | Discrete geometry Diskrete Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020537199&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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