Cone Depth and the Center Vertex Theorem

We generalize the Tukey depth to use cones instead of halfspaces. We prove a generalization of the center point theorem that for SRd, there is a point s 2 S, with depth at least n d+1 for cones of half-angle 45 ◦ . This gives a notion of data depth for which an approximate median can always be found among the original set.