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What is the combinatorial complexity of the Voronoi diagram of a set of lines (or line segments) in three dimensions?
Uncertain, pending investigation.
Open. Conjectured to be nearly quadradic.
There is a gap between a lower bound of \Omega(n^2) and an upper bound that is essentially cubic [Sha94] for the Euclidean case (and yet is quadratic for polyhedral metrics [BSTY98]). A recent advance shows that the “level sets” of the Voronoi diagram of lines, given by the union of a set of cylinders, indeed has near-quadratic complexity [AS00b].
This problem is closely related to Problem 2, because points moving in the plane with constant velocity yield straight-line trajectories in space-time.
Voronoi diagrams
J. O’Rourke, 2 Aug. 2001; 13 Dec. 2001.
Pankaj K. Agarwal and Micha Sharir. Pipes, cigars, and kreplach: The union of Minkowski sums in three dimensions. Discrete Comput. Geom., 24(4):645–685, 2000.
Jean-Daniel Boissonnat, Micha Sharir, Boaz Tagansky, and Mariette Yvinec. Voronoi diagrams in higher dimensions under certain polyhedral distance functions. Discrete Comput. Geom., 19(4):473–484, 1998.
J. S. B. Mitchell and Joseph O’Rourke. Computational geometry column 42. Internat. J. Comput. Geom. Appl., 11(5):573–582, 2001. Also in SIGACT News 32(3):63-72 (2001), Issue 120.
Micha Sharir. Almost tight upper bounds for lower envelopes in higher dimensions. Discrete Comput. Geom., 12:327–345, 1994.