. We show that, for any set of n points in d dimensions, there exists a hyperplane 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 dimensions there exists a point that cannot escape to infinity without crossing 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 computational complexity of regression depth problems. 1. Introduction Robust statistics [13], [32] has attracted much attention recently within the computational geometry community due to the natural geometric formulation of many of its problems. In contrast to least-squares regression, in which measurement error is assumed to be # The work of Amenta, Eppstein, and Teng was performed in part while visiting the Xerox Palo Alto Research Center. David Eppstein'...
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