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Let $S$ be a set of lines (or line segments) in $\mathbb{R}^3$. Consider the Voronoi diagram $VD(S)$. The best lower bound on the complexity of $VD(S)$ is $\Omega(n^2)$ and the best upper bound is $O(n^{3+\epsilon})$ [1]. What is the current state of research on this problem? Is anything more recent than the references in [2] known?

[1] Almost tight upper bounds for lower envelopes in higher dimensions.

[2] http://cs.smith.edu/~orourke/TOPP/P3.html

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In 2010 Hemmer et al. gave an exact algorithm for the Voronoi diagram of lines in $\mathbb{R}^3$. In their introduction, they state that the $\Omega(n^2)$ lower bound and $O(n^{3+\varepsilon})$ upper bound are still the best known on the combinatorial complexity of the resulting Voronoi diagram. Their algorithm runs in time $O(n^{3+\varepsilon})$, matching the upper bound on the combinatorial complexity.

As far as I know, theirs was the first (and still, only) exact algorithm for this problem, so I would guess that there haven't been any further improvements on the combinatorial complexity front either.

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The problem is still open, although some progress was made on related problems (see http://arxiv.org/abs/1312.2194). It is known that if you are willing to use an approximate metric then the upper bound drops to near quadratic. See the work by Koltun and Sharir, I think.

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