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.