Lectures on Discrete Geometry

Front Cover
Springer Science & Business Media, Dec 1, 2013 - Mathematics - 486 pages
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.
 

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Contents

Lattices and Minkowskis Theorem
17
Convex Independent Subsets
29
Incidence Problems
41
Convex Polytopes
77
Number of Faces in Arrangements
125
Lower Envelopes 165
164
Intersection Patterns of Convex Sets
195
Geometric Selection Theorems
207
Two Applications of HighDimensional Polytopes
289
Volumes in High Dimension
311
Measure Concentration and Almost Spherical Sections
329
Embedding Finite Metric Spaces into Normed Spaces 355
354
What Was It About? An Informal Summary
401
Hints to Selected Exercises
409
Bibliography
417
Index 459
458

Transversals and Epsilon Nets
231
Attempts to Count kSets 265
264

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