Better Late than Never: the Complexity of Arrangements of Polyhedra

2025-06-04

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.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningSearch and Optimization: Combinatorial OptimizationConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphsSecurity and Privacy: Large-scale security measurementsSemantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data

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

• Analyzing combinatorial complexity of convex polyhedra arrangements
• Proving tight bounds for d-dimensional polyhedral subdivisions
• Addressing unproven literature claims on arrangement complexity

Innovation

• Proves tight combinatorial complexity bounds
• Analyzes convex polyhedra arrangements
• Generalizes to arbitrary dimensions