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 \(\varphi _{\mu }(x)=\inf \{\mu (H):H\in {{{\mathcal {H}}}}(x)\}\), where \(\mathcal{H}(x)\) is the set of all half-spaces H of \({{\mathbb {R}}}^n\) containing x. We show that if \(\mu \) is a non-degenerate log-concave probability measure on \({{\mathbb {R}}}^n\) then

$$\begin{aligned} e^{-c_1n}\leqslant \int _{{\mathbb {R}}^n}\varphi _{\mu }(x)\,d\mu (x) \leqslant e^{-c_2n/L_{\mu }^2} \end{aligned}$$

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.