Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, October 17, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Patrick Schnider

Bisecting three classes of lines

Consider the following problem: Let L be an arrangement of n lines in R^3 colored red, green, and blue. Does there exist a vertical plane P such that a line on P simultaneously bisects all three classes of points in the cross-section given by the intersection of L and P? In a previous talk at the Mittagsseminar I showed the existence of such a plane using topological methods. In this talk, we will discuss an alternative proof of this fact, using only methods from discrete geometry. With this combinatorial proof at hand, we devise an O(n^2 log^2(n)) time algorithm to find such a plane and the bisector of the induced cross-section. We do this by providing a general framework, which can be applied to solve similar problems on cross-sections and kinetic points. This is joint work with Alexander Pilz.

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