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

Let x1,…,xNx1,…,xN be independent random points distributed according to an isotropic log-concave measure μ   on RnRn, and consider the random polytopeKN:=conv{±x1,…,±xN}.KN:=conv{±x1,…,±xN}. We provide sharp estimates for the quermaßintegrals and other geometric parameters of KNKN in the range cn⩽N⩽exp⁡(n)cn⩽N⩽exp⁡(n); these complement previous results from [13] and [14] that were given for the range cn⩽N⩽exp⁡(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 Zq(μ)Zq(μ) of μ   for all 1⩽q⩽n1⩽q⩽n.