Cone Depth and the Center Vertex Theorem

Presented at The Fall Workshop in in Computational Geometry, October 31, 2008

We generalize the Tukey depth to use cones instead of halfspaces.
We prove a generalization of the Center Point Theorem that for $S\subset \reals^d$,
there is a vertex $s\in S$, with depth at least $\frac{n}{d+1}$ for cones of half-angle
$45^\circ$. This gives a notion of data depth for which an approximate median can always
be found among the original set. We present a simple algorithm for computing an
approximate center vertex.