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

Date and Time: Tuesday, April 04, 2023, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Patrick Schnider

Tverberg's Theorem for Hyperplanes

Tverberg's theorem is a fundamental theorem in discrete geometry which implies many other famous results such as Radons lemma and the Centerpoint theorem. It states that any r(d-1)+1 points in d-dimensional Euclidean space can be partitioned into r parts whose convex hulls have a common intersection. Motivated by an application in statistics, Rousseeuw and Hubert conjectured in 1999 that an analogue statement should be true when replacing "points" with "hyperplanes" and "convex hull" with "positive regression depth" (which will be defined in the talk). I will outline the motivation of the conjecture of Rousseeuw and Hubert by introducing combinatorial depth measures for hyperplane arrangements and show that the conjecture is true. This is joint work with Pablo Soberon.

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