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 $V(z)/\text{ Vol }(B)$ , where $V(z)$ is the minimum volume of the intersection of $B$ with the halfspaces defined by any hyperplane passing through $z$ , and $\text{ Vol }(B)$ 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.