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 xin {{mathbb {R}}}^n by varphi _{mu }(x)=inf {mu (H):Hin {{{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 e-c1n⩽∫Rnφμ(x)dμ(x)⩽e-c2n/Lμ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} e^{-c_1n}\\leqslant \\int _{{\\mathbb {R}}^n}\\varphi _{\\mu }(x)\\,d\\mu (x) \\leqslant e^{-c_2n/L_{\\mu }^2} \\end{aligned}$$\\end{document}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.