The isotropic constant of random polytopes with vertices on convex surfaces.

For an isotropic convex body K⊂Rn we consider the isotropic constant LKN of the symmetric random polytope KN generated by N independent random points which are distributed according to the cone probability measure on the boundary of K. We show that with overwhelming probability LKN≤Clog(2N∕n), where C∈(0,∞) is an absolute constant. If K is unconditional we argue that even LKN≤C with overwhelming probability and thereby verify the hyperplane conjecture for this model. The proofs are based on concentration inequalities for sums of sub-exponential or sub-Gaussian random variables, respectively, and, in the unconditional case, on a new ψ2-estimate for linear functionals with respect to the cone measure in the spirit of Bobkov and Nazarov, which might be of independent interest.