Regression depth and center points

We show that, for any set of n points in d dimensions, there exists a hyper plane with regression depth at least [n/(d+ 1)], as had been conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n hyperplanes in d di mensions there exists a point that cannot escape to infinity without,crossi ng at least [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 com putational complexity of regression depth problems.