Inscribable Fans II: Inscribed zonotopes, simplicial arrangements, and reflection groups

Abstract

An arrangement of hyperplanes is strongly inscribable if it has an inscribed (or ideal hyperbolic) zonotope. We characterize inscribed zonotopes and prove that the family of strongly inscribable arrangements is closed under restriction and localization. Moreover, we show that (strongly) inscribable arrangements are simplicial. We conjecture that only reflection arrangements and their restrictions are strongly inscribable and we verify our conjecture in rank-$3$ using the conjecturally complete list of irreducible simplicial rank-$3$ arrangements.