Better Late than Never: the Complexity of Arrangements of Polyhedra

TL;DR

This paper establishes tight bounds on the combinatorial complexity of arrangements of convex polyhedra in arbitrary dimensions, resolving a long-standing open problem in computational geometry.

Contribution

It provides the first published proof of the tight complexity bounds for arrangements of convex polyhedra in any dimension.

Findings

Complexity bound of O(m^{ceil(d/2)} n^{floor(d/2)}) established

Proof of tightness of the complexity bound provided

Addresses a previously unresolved problem in geometric combinatorics

Abstract

Let $\mathcal{A}$ be the subdivision of $\mathbb{R}^d$ induced by $m$ convex polyhedra having $n$ facets in total. We prove that $\mathcal{A}$ has combinatorial complexity $O(m^{\lceil d/2 \rceil} n^{\lfloor d/2 \rfloor})$ and that this bound is tight. The bound is mentioned several times in the literature, but no proof for arbitrary dimension has been published before.