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arXiv · 2506.03960

Better Late than Never: the Complexity of Arrangements of Polyhedra

Boris Aronov·Sang Won Bae·Sergio Cabello·Otfried Cheong·David Eppstein·Christian Knauer·Raimund Seidel
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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.

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Boris Aronov, Sang Won Bae, Sergio Cabello, Otfried Cheong, David Eppstein, Christian Knauer, Raimund Seidel. 2025-06-04. Better Late than Never: the Complexity of Arrangements of Polyhedra. https://arxiv.org/abs/2506.03960

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