Half-space depth of log-concave probability measures

Abstract

Given a probability measure $\mu$ on ${\mathbb{R}}^n$, Tukey's half-space depth is defined for any $x\in {\mathbb{R}}^n$ by ${\phi }_{\mu }(x)=\inf \{\mu (H):H\in \mathcal{H}(x)\}$, where $\mathcal{H}(x)$ is the set of all half-spaces $H$ of ${\mathbb{R}}^n$ containing $x$. We show that if $\mu$ is log-concave then $$e^{-c_{1}n}\le {\int }_{{\mathbb{R}}^n}{\phi }_{\mu }(x)d\mu (x)\le e^{-c_{2}n/L_{\mu }^2}$$ where $L_{\mu }$ is the isotropic constant of $\mu$ and $c_1,c_2>0$ are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of $L_q$-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.

arXiv commentFinal version, to appear in Probability Theory and Related Fields