The complexity of the envelope of line and plane arrangements

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 $(n-4)(2n-3)+5$.