We show that, for any set of n points in d dimensions, there exists a hyperplane with regression depth at least ceiling(n/(d+1)), as had been conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n hyperplanes in d dimensions there exists a point that cannot escape to infinity without crossing at least ceiling(n/(d+1)) hyperplanes. We also apply our approach to related questions on the existence of partitions of the data into subsets such that a common plane has nonzero regression depth in each subset, and to the computational complexity of regression depth problems.
This is joint work with Nina Amenta, Marshall Bern and Shang-Hua Teng.