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)).

Let C be a symmetric convex body of volume 1 in ${\mathbb R}^n$. We provide general estimates for the volume and the radius of C ∩ U(C) where U is a random orthogonal transformation of ${\mathbb R}^n$. In particular, we consider the case where C is in the isotropic position or C is the volume normalized Lq-centroid body Zq(μ) of an isotropic log-concave measure μ on ${\mathbb R}^n$.

No abstract

No abstract

No abstract