About the number of connected components in arrangements of hyperplanes

TL;DR

This paper investigates the maximum number of connected components in arrangements of hyperplanes in real projective spaces and shows that for large n and dimensions greater than 3, most numbers between n and this maximum are realizable.

Contribution

It extends known results for dimensions 2 and 3 to higher dimensions, demonstrating that almost all intermediate counts are achievable in large hyperplane arrangements.

Findings

Most integers between n and the maximum number of regions are realizable for large n and d>3.

The maximum number of regions is well-characterized for hyperplane arrangements.

The result generalizes previous findings from lower dimensions to higher dimensions.

Abstract

We consider arrangements of n hyperplanes of codimension one in a real projective space of dimension d. Let us denote by F the maximal possible number f of connected components of the complement in the projective space of dimension d to the union of n hyperplanes. We prove that for sufficiently large n and for d>3 almost all integers between n and F could be realized as the numbers of regions for some arrangements of n hyperplanes in the projective space. This fact was known before for d=2,3.