Half-space depth of log-concave probability measures

Abstract

Abstract Given a probability measure $$\mu $$ μ on $${{\mathbb {R}}}^n$$ R n , Tukey’s half-space depth is defined for any $$x\in {{\mathbb {R}}}^n$$ x ∈ R n by $$\varphi _{\mu }(x)=\inf \{\mu (H):H\in {{{\mathcal {H}}}}(x)\}$$ φ μ ( x ) = inf { μ ( H ) : H ∈ H ( x ) } , where $$\mathcal{H}(x)$$ H ( x ) is the set of all half-spaces H of $${{\mathbb {R}}}^n$$ R n containing x . We show that if $$\mu $$ μ is a non-degenerate log-concave probability measure on $${{\mathbb {R}}}^n$$ R n then $$\begin{aligned} e^{-c_1n}\leqslant \int _{{\mathbb {R}}^n}\varphi _{\mu }(x)\,d\mu (x) \leqslant e^{-c_2n/L_{\mu }^2} \end{aligned}$$ e - c 1 n ⩽ ∫ R n φ μ ( x ) d μ ( x ) ⩽ e - c 2 n / L μ 2 where $$L_{\mu }$$ L μ is the isotropic constant of $$\mu $$ μ and $$c_1,c_2>0$$ c 1 , c 2 > 0 are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of $$L_q$$ 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.