Remarks on the Relation of Log-Concave and Contoured Distributions in \({\mathbb {R}}^n\)

We provide a characterization of those log-concave distributions in \({{\mathbb {R}}}^n\) ${\mathbb{R}}^n$ that are contoured distributions, through the \(K_p\) $K_p$ -bodies of the distribution, defined by Ball. Our method uses the logarithmic integral for the solution of a Bernstein type approximation problem. In the second part of the paper we state a question for contoured distributions that would provide an alternative approach to the isotropic constant problem.