Sharpening work of the first two authors, for every proportion ∈ (0, 1) we provide exact quantitative relations between global parameters of n-dimensional symmetric convex bodies and the diameter of their random n -dimensional sections. Using recent results of Gromov and Vershynin, we obtain an "asymptotic formula" for the diameter of random proportional sections.
We study some geometric properties of the L q -centroid bodies Z q (μ) of an isotropic log-concave measure μ on R n . For any 2 q √ n and for ε ∈ (ε 0 (q, n), 1) we determine the inradius of a random (1−ε)n-dimensional projection of Z q (μ) up to a constant depending polynomially on ε. Using this fact we obtain estimates for the covering numbers N ( √ qB n 2 , tZ q (μ)), t 1, thus showing that Z q (μ) is a β-regular convex body. As a consequence, we also get an upper bound for M (Z q (μ)).
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