Asymptotic shape of the convex hull of isotropic log-concave random vectors

Let $x_1,\ldots ,x_N$ be independent random points distributed according to an isotropic log-concave measure $\mu $ on ${\mathbb R}^n$, and consider the random polytope $$K_N:={\rm conv}\{ \pm x_1,\ldots ,\pm x_N\}.$$ We provide sharp estimates for the querma\ss{}integrals and other geometric parameters of $K_N$ in the range $cn\ls N\ls\exp (n)$; these complement previous results from \cite{DGT1} and \cite{DGT} that were given for the range $cn\ls N\ls\exp (\sqrt{n})$. One of the basic new ingredients in our work is a recent result of E.~Milman that determines the mean width of the centroid body $Z_q(\mu )$ of $\mu $ for all $1\ls q\ls n$.