Isotropic constant of random polytopes with vertices on an $\ell_p$-sphere

Julia Hörrmann, Joscha Prochno, Christoph Thaele

Published 2016-05-30Version 1

The symmetric convex hull of random points that are independent and distributed according to the cone measure on the unit sphere of $\ell_p^n$ for some $1\leq p < \infty$ is considered. We prove that these random polytopes have uniformly absolutely bounded isotropic constants with overwhelming probability. This generalizes the result for the Euclidean sphere ($p=2$) obtained by D. Alonso-Guti\'errez. The proof requires several different tools including a concentration inequality for the cone measure due to G. Schechtman and J. Zinn and moment estimates for sums of independent random variables with log-concave tails originating in the work of E. Gluskin and S. Kwapie\'n.