The complexity of the envelope of line and plane arrangements

TL;DR

This paper investigates the minimal number of external facets in hyperplane arrangements, establishing exact and bounded values for arrangements of lines and planes, thus advancing understanding of their combinatorial complexity.

Contribution

It provides new exact and bounded formulas for the minimum number of external facets in simple arrangements of lines and planes.

Findings

Minimum external facets for arrangements of 4+ lines is 2(n-1).

Minimum external facets for arrangements of 5+ planes is between (n(n-2)+6)/3 and (n-4)(2n-3)+5.

The results refine previous hypotheses on the complexity of arrangement envelopes.

Abstract

A facet of an hyperplane arrangement is called external if it belongs to exactly one bounded cell. The set of all external facets forms the envelope of the arrangement. The number of external facets of a simple arrangement defined by $n$ hyperplanes in dimension $d$ is hypothesized to be at least $d{n-2 \choose d-1}$. In this note we show that, for simple arrangements of 4 lines or more, the minimum number of external facets is equal to $2(n-1)$, and for simple arrangements of 5 planes or more, the minimum number of external facets is between $\frac{n(n-2)+6}{3}$ and $(n-4)(2n-3)+5$.