Oja centers and centers of gravity

Oja depth (Oja 1983) is a generalization of the median to multivariate data that measures the centrality of a point x with respect to a set S of points in such a way that points with smaller Oja depth are more central with respect to S. Two relationships involving Oja depth and centers of mass are presented. The first is a form of Centerpoint Theorem which shows that the center of mass of the convex hull of a point set has low Oja depth. The second is an approximation result which shows that the center of mass of a point set approximates a point of minimum Oja depth. This work was initiated during the Workshop on Computational Geometry 2006 in Caldes de Malavella. The authors wish to thank Ferran Hurtado and the organizers for the opportunity of working on this topic. Email addresses: [email protected] (Dan Chen), [email protected] (Olivier Devillers), [email protected] (John Iacono), [email protected] (Stefan Langerman), [email protected] ( Pat Morin) Preprint submitted to Elsevier February 28, 2012