Better Late than Never: the Complexity of Arrangements of Polyhedra
🤖 AI Summary
This paper investigates the combinatorial complexity of the arrangement induced by $m$ convex polyhedra with a total of $n$ faces in $mathbb{R}^d$. We establish, for the first time, the tight asymptotic bound $O(m^{lceil d/2
ceil} n^{lfloor d/2
floor})$, resolving a long-standing open problem wherein this bound had been widely cited but never rigorously proven. Our approach integrates tools from combinatorial geometry, hierarchical projection analysis, duality transformations, and inductive construction to derive an asymptotically precise characterization valid in all dimensions. Furthermore, we provide an explicit family of constructions achieving this bound, thereby confirming its tightness. This result fills a fundamental gap in the theory of high-dimensional polyhedral arrangements and furnishes foundational complexity guarantees for applications in computational geometry, geometric optimization, and spatial partitioning.
Application Category
📝 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 $O(m^{lceil d/2
ceil} n^{lfloor d/2
floor})$ 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.
Problem
Research questions and friction points this paper is trying to address.
Analyzing combinatorial complexity of convex polyhedra arrangements
Proving tight bounds for d-dimensional polyhedral subdivisions
Addressing unproven literature claims on arrangement complexity
Innovation
Methods, ideas, or system contributions that make the work stand out.
Proves tight combinatorial complexity bounds
Analyzes convex polyhedra arrangements
Generalizes to arbitrary dimensions
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