Voronoi diagrams of moving points in the plane and of lines in space: tight bounds for simple configurations

The combinatorial complexities of (1) the Voronoi diagram of moving points in 2D and (2) the Voronoi diagram of lines in 3D, both under the Euclidean metric, continues to challenge geometers because of the open gap between the Ω(n2) lower bound and the O(n3+ɛ) upper bound. Each of these two combinatorial problems has a closely related problem involving Minkowski sums: (1′) the complexity of a Minkowski sum of a planar disk with a set of lines in 3D and (2′) the complexity of a Minkowski sum of a sphere with a set of lines in 3D. These Minkowski sums can be considered “cross-sections” of the corresponding Voronoi diagrams. Of the four complexity problems mentioned, problems (1′) and (2′) have recently been shown to have a nearly tight bound: both complexities are O(n2+ɛ) with lower bound Ω(n2).