Half-space depth of log-concave probability measures.

Given a probability measure μ on R n , Tukey's half-space depth is defined for any x ∈ R n by φ μ (x) = inf { μ (H) : H ∈ H (x) } , where H (x) is the set of all half-spaces H of R n containing x. We show that if μ is a non-degenerate log-concave probability measure on R n then e - c 1 n ⩽ ∫ R n φ μ (x) d μ (x) ⩽ e - c 2 n / L μ 2 where L μ is the isotropic constant of μ 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. [ABSTRACT FROM AUTHOR]

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