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${\cal H}(x)$isthesetofallhalf-spaces$H$of${\mathbb R}^n$containing$x$\mathrm{.}Weshowthatif$\mu $islog-concavethen$$e^{-c_1n}\leq \int_{$\mathbb{R}$^n}\varphi_{\mu }(x),d\mu(x) \leq e^{-c_2n/L_{\mu}^2}$$where$L_{\mu }$istheisotropicconstantof$\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.