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离散几何讲义PDF|Epub|txt|kindle电子书版本网盘下载
- （捷克）马陶塞克著 著
- 出版社： 北京：世界图书北京出版公司
- ISBN：9787510037627
- 出版时间：2011
- 标注页数：481页
- 文件大小：90MB
- 文件页数：502页
- 主题词：离散数学：几何学－英文
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图书目录
1 Convexity1
1.1 Linear and Affine Subspaces,General Position1
1.2 Convex Sets,Convex Combinations,Separation5
1.3 Radon's Lemma and Helly's Theorem9
1.4 Centerpoint and Ham Sandwich14
2 Lattices and Minkowski's Theorem17
2.1 Minkowski's Theorem17
2.2 General Lattices21
2.3 An Application in Number Theory27
3 Convex Independent Subsets29
3.1 The Erd？s-Szekeres Theorem30
3.2 Horton Sets34
4 Incidence Problems41
4.1 Formulation41
4.2 Lower Bounds:Incidences and Unit Distances51
4.3 Point-Line Incidences via Crossing Numbers54
4.4 Distinct Distances via Crossing Numbers59
4.5 Point-Line Incidences via Cuttings64
4.6 A Weaker Cutting Lemma70
4.7 The Cutting Lemma:A Tight Bound73
5 Convex Polytopes77
5.1 Geometric Duality78
5.2 H-Polytopes and V-Polytopes82
5.3 Faces of a Convex Polytope86
5.4 Many Faces:The Cyclic Polytopes96
5.5 The Upper Bound Theorem100
5.6 The Gale Transform107
5.7 Voronoi Diagrams115
6 Number of Faces in Arrangements125
6.1 Arrangements of Hyperplanes126
6.2 Arrangements of Other Geometric Objects130
6.3 Number of Vertices of Level at Most k140
6.4 The Zone Theorem146
6.5 The Cutting Lemma Revisited152
7 Lower Envelopes165
7.1 Segments and Davenport-Schinzel Sequences165
7.2 Segments:Superlinear Complexity of the Lower Envelope169
7.3 More on Davenport-Schinzel Sequences173
7.4 Towards the Tight Upper Bound for Segments178
7.5 Up to Higher Dimension:Triangles in Space182
7.6 Curves in the Plane186
7.7 Algebraic Surface Patches189
8 Intersection Patterns of Convex Sets195
8.1 The Fractional Helly Theorem195
8.2 The Colorful Carathéodory Theorem198
8.3 Tverberg's Theorem200
9 Geometric Selection Theorems207
9.1 A Point in Many Simplices:The First Selection Lemma207
9.2 The Second Selection Lemma210
9.3 Order Types and the Same-Type Lemma215
9.4 A Hypergraph Regularity Lemma223
9.5 A Positive-Fraction Selection Lemma228
10 Transversals and Epsilon Nets231
10.1 General Preliminaries:Transversals and Matchings231
10.2 Epsilon Nets and VC-Dimension237
10.3 Bounding the VC-Dimension and Applications243
10.4 Weak Epsilon Nets for Convex Sets251
10.5 The Hadwiger-Debrunner(p,q)-Problem255
10.6 A(p,q)-Theorem for Hyperplane Transversals259
11 Attempts to Count k-Sets265
11.1 Definitions and First Estimates265
11.2 Sets with Many Halving Edges273
11.3 The Lovász Lemma and Upper Bounds in All Dimensions277
11.4 A Better Upper Bound in the Plane283
12 Two Applications of High-Dimensional Polytopes289
12.1 The Weak Perfect Graph Conjecture290
12.2 The Brunn-Minkowski Inequality296
12.3 Sorting Partially Ordered Sets302
13 Volumes in High Dimension311
13.1 Volumes,Paradoxes of High Dimension,and Nets311
13.2 Hardness of Volume Approximation315
13.3 Constructing Polytopes of Large Volume322
13.4 Approximating Convex Bodies by Ellipsoids324
14 Measure Concentration and Almost Spherical Sections329
14.1 Measure Concentration on the Sphere330
14.2 Isoperimetric Inequalities and More on Concentration333
14.3 Concentration of Lipschitz Functions337
14.4 Almost Spherical Sections:The First Steps341
14.5 Many Faces of Symmetric Polytopes347
14.6 Dvoretzky's Theorem348
15 Embedding Finite Metric Spaces into Normed Spaces355
15.1 Introduction:Approximate Embeddings355
15.2 The Johnson-Lindenstrauss Flattening Lemma358
15.3 Lower Bounds By Counting362
15.4 A Lower Bound for the Hamming Cube369
15.5 A Tight Lower Bound via Expanders373
15.6 Upper Bounds for ？∞-Embeddings385
15.7 Upper Bounds for Euclidean Embeddings389
What Was It About?An Informal Summary401
Hints to Selected Exercises409
Bibliography417
Index459