Lower bound on the Voronoi diagram of lines in $\mathbb{R}^d$

TL;DR

This paper establishes a fundamental lower bound on the complexity of Euclidean Voronoi diagrams for non-intersecting lines in higher-dimensional spaces, revealing inherent geometric limitations.

Contribution

It provides the first known lower bound of (n^{2d/3}) for the maximal complexity of Voronoi diagrams of lines in () dimensions.

Findings

Lower bound of (n^{2d/3}) for Voronoi diagram complexity

Complexity grows polynomially with the number of lines and dimension

Results apply to non-intersecting lines in () space

Abstract

This note gives a lower bound of $\Omega(n^{\lceil 2d/3\rceil})$ on the maximal complexity of the Euclidean Voronoi diagram of $n$ non-intersecting lines in $\mathbb{R}^d$ for $d>2$.