Better Late than Never: the Complexity of Arrangements of Polyhedra

TL;DR

The paper resolves the combinatorial complexity of the arrangement induced by $m$ convex polyhedra with total $n$ facets in $\mathbb{R}^d$, proving the tight bound $O(m^{\lceil d/2 \rceil} n^{\lfloor d/2 \rfloor})$ on the number of faces. It develops a Seidel-style counting argument in general position, extends it to arbitrary polyhedra via a perturbation scheme that preserves face counts inside a bounding region, and provides a matching lower bound through a product-polytope construction. This unifies extreme cases (single polyhedron and hyperplane arrangements) and clarifies a bound long cited in the literature. The result has implications for understanding the intrinsic complexity of polyhedral arrangements in fixed dimension and informs related computational-geometric analyses.

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.

Theorems & Definitions (2)

• Theorem 1
• Lemma 2