Half-space depth of log-concave probability measures

Given a probability measure $\mu $ on ${\mathbb R}^n$, Tukey's half-space depth is defined for any $x\in {\mathbb R}^n$ by $\varphi_{\mu }(x)=\inf\{\mu (H):H\in {\cal H}(x)\}$, where ${\cal H}(x)$ is the set of all half-spaces $H$ of ${\mathbb R}^n$ containing $x$. We show that if $\mu $ is log-concave then $$e^{-c_1n}\leq \int_{\mathbb{R}^n}\varphi_{\mu }(x)\,d\mu(x) \leq e^{-c_2n/L_{\mu}^2}$$ 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