Data depth for the uniform distribution

Given a set X of k points and a point z in the n-dimensional euclidean space, the Tukey depth of z with respect to X, is defined as m/k, where m is the minimum integer such that z is not in the convex hull of some set of k-m points of X. If z belongs to the closed region B delimited by an ellipsoid, define the continuous depth of z with respect to B as the quotient V(z)/Vol(B), where V(z) is the minimum volume of the intersection of B with the halfspaces defined by any hyperplane passing through z, and Vol(B) is the volume of B. We consider z a random variable and prove that, if z is uniformly distributed in B, the continuous depth of z with respect to B has expected value 1/2^n+1. This result implies that if z and X are uniformly distributed in B, the expected value of Tukey depth of z with respect to X converges to 1/2^n+1 as the number of points k goes to infinity. These findings have applications in ecology, namely within the niche theory, where it is useful to explore and characterize the distribution of points inside species niche.