Better Late than Never: the Complexity of Arrangements of Polyhedra

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 $O(m^{\lceil d\mathrm{/}2\rceil }{n}^{\lfloor d\mathrm{/}2\rfloor })$ 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.