The isotropic constant of random polytopes with vertices on convex surfaces

For an isotropic convex body K subset of R-n we consider the isotropic constant L-KN of the symmetric random polytope K-N 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 L-KN <= C root log(2N/n), where C is an element of(0, infinity) is an absolute constant. If K is unconditional we argue that even L-KN <= 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 psi(2)-estimate for linear functionals with respect to the cone measure in the spirit of Bobkov and Nazarov, which might be of independent interest. (C) 2019 Elsevier Inc. All rights reserved.