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

TL;DR

This paper characterizes inscribed zonotopes and strongly inscribable hyperplane arrangements, proving their properties and conjecturing their relation to reflection arrangements, with verification in rank-3 cases.

Contribution

It provides a characterization of inscribed zonotopes, proves closure properties of strongly inscribable arrangements, and conjectures their connection to reflection arrangements, verified in rank-3.

Findings

Strongly inscribable arrangements are closed under restriction and localization.

Inscribable arrangements are necessarily simplicial.

Conjecture verified for rank-3 arrangements.

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.