Lectures on Discrete Geometry,9780387953748

Lectures on Discrete Geometry

by
ISBN13: 9780387953748
ISBN10: 0387953744
Format: Paperback
Pub. Date: 2002-07-01
Publisher(s): Springer Nature
List Price: $83.99

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Summary

This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces.

Table of Contents

Preface v
Notation and Terminology xv
Convexity
1(16)
Linear and Affine Subspaces, General Position
1(4)
Convex Sets, Convex Combinations, Separation
5(4)
Radon's Lemma and Helly's Theorem
9(5)
Centerpoint and Ham Sandwich
14(3)
Lattices and Minkowski's Theorem
17(12)
Minkowski's Theorem
17(4)
General Lattices
21(6)
An Application in Number Theory
27(2)
Convex Independent Subsets
29(12)
The Erdos--Szekeres Theorem
30(4)
Horton Sets
34(7)
Incidence Problems
41(36)
Formulation
41(10)
Lower Bounds: Incidences and Unit Distances
51(3)
Point--Line Incidences via Crossing Numbers
54(5)
Distinct Distances via Crossing Numbers
59(5)
Point--Line Incidences via Cuttings
64(6)
A Weaker Cutting Lemma
70(3)
The Cutting Lemma: A Tight Bound
73(4)
Convex Polytopes
77(48)
Geometric Duality
78(4)
H-Polytopes and V-Polytopes
82(4)
Faces of a Convex Polytope
86(10)
Many Faces: The Cyclic Polytopes
96(4)
The Upper Bound Theorem
100(7)
The Gale Transform
107(8)
Voronoi Diagrams
115(10)
Number of Faces in Arrangements
125(40)
Arrangements of Hyperplanes
126(4)
Arrangements of Other Geometric Objects
130(10)
Number of Vertices of Level at Most k
140(6)
The Zone Theorem
146(6)
The Cutting Lemma Revisited
152(13)
Lower Envelopes
165(30)
Segments and Davenport-Schinzel Sequences
165(4)
Segments: Superlinear Complexity of the Lower Envelope
169(4)
More on Davenport-Schinzel Sequences
173(5)
Towards the Tight Upper Bound for Segments
178(4)
Up to Higher Dimension: Triangles in Space
182(4)
Curves in the Plane
186(3)
Algebraic Surface Patches
189(6)
Intersection Patterns of Convex Sets
195(12)
The Fractional Helly Theorem
195(3)
The Colorful Caratheodory Theorem
198(2)
Tverberg's Theorem
200(7)
Geometric Selection Theorems
207(24)
A Point in Many Simplices: The First Selection Lemma
207(3)
The Second Selection Lemma
210(5)
Order Types and the Same-Type Lemma
215(8)
A Hypergraph Regularity Lemma
223(5)
A Positive-Fraction Selection Lemma
228(3)
Transversals and Epsilon Nets
231(34)
General Preliminaries: Transversals and Matchings
231(6)
Epsilon Nets and VC-Dimension
237(6)
Bounding the VC-Dimension and Applications
243(8)
Weak Epsilon Nets for Convex Sets
251(4)
The Hadwiger--Debrunner (p, q)-Problem
255(4)
A (p, q)-Theorem for Hyperplane Transversals
259(6)
Attempts to Count k-Sets
265(24)
Definitions and First Estimates
265(8)
Sets with Many Halving Edges
273(4)
The Lovasz Lemma and Upper Bounds in All Dimensions
277(6)
A Better Upper Bound in the Plane
283(6)
Two Applications of High-Dimensional Polytopes
289(22)
The Weak Perfect Graph Conjecture
290(6)
The Brunn--Minkowski Inequality
296(6)
Sorting Partially Ordered Sets
302(9)
Volumes in High Dimension
311(18)
Volumes, Paradoxes of High Dimension, and Nets
311(4)
Hardness of Volume Approximation
315(7)
Constructing Polytopes of Large Volume
322(2)
Approximating Convex Bodies by Ellipsoids
324(5)
Measure Concentration and Almost Spherical Sections
329(26)
Measure Concentration on the Sphere
330(3)
Isoperimetric Inequalities and More on Concentration
333(4)
Concentration of Lipschitz Functions
337(4)
Almost Spherical Sections: The First Steps
341(6)
Many Faces of Symmetric Polytopes
347(1)
Dvoretzky's Theorem
348(7)
Embedding Finite Metric Spaces into Normed Spaces
355(46)
Introduction: Approximate Embeddings
355(3)
The Johnson-Lindenstrauss Flattening Lemma
358(4)
Lower Bounds By Counting
362(7)
A Lower Bound for the Hamming Cube
369(4)
A Tight Lower Bound via Expanders
373(12)
Upper Bounds for l∞-Embeddings
385(4)
Upper Bounds for Euclidean Embeddings
389(12)
What Was It About? An Informal Summary 401(8)
Hints to Selected Exercises 409(8)
Bibliography 417(42)
Index 459

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