Fundamental Concepts in Combinatorial Geometry

Abstract

Two of the main subjects studied in combinatorial geometry and therefore in this book are finite sets of points and finite sets of hyperplanes. Not all questions about finite sets of points or hyperplanes are combinatorial, though, and one has to keep in mind that a strict classification into combinatorial and non-combinatorial problems is neither reasonable nor desirable. Nevertheless, there are a few characteristics that identify a problem as combinatorial. For example, a typical combinatorial question that can be asked about a set P of n points in d-dimensional Euclidean space E d is the following: “How many partitions of P into two subsets can be defined by hyperplanes?” If H is a set of n hyperplanes in the same space, then it is a combinatorial question if one asks “What is the number of cells the space is cut into by the hyperplanes in H?” We will investigate both problems and many related ones in this chapter and, more generally, in this book.