Semispaces of Configurations

Abstract

A non-vertical hyperplane h, disjoint from a finite set P of points, partitions P into two sets P⁺ = P∩h⁺ and P^- = P∩h^- called semispaces of P, where h⁺ is the open half-space above hyperplane h and h^- is the open half-space below h. Note that this definition includes the empty set and P itself as semispaces of P. Using geometric transformations and the face counting formulas for arrangements of hyper planes presented in Chapter 1, it is not hard to find tight upper bounds on the number of semispaces of P that depend solely on the cardinality of P. Unfortunately, little is known about the maximum number of semispaces with some fixed cardinality. This chapter addresses the latter counting problem and derives non-trivial upper and lower bounds.