On the isotropic constant of random polytopes with vertices on an ℓp-Sphere - Hörrmann, Julia; Prochno, Joscha; Thäle, Christoph (The Journal of Geometric Analysis, 2018)
The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the p-unit sphere of Rn for some 1 ≤ p < ∞ 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 Alonso-Gutiérrez. The proof requires several different tools including a probabilistic representation of the cone measure due to Schechtman and Zinn and moment estimates for sums of independent random variables with log-concave tails originating in a paper of Gluskin and Kwapien.
Keywords: Asymptotic convex geometry Cone measure Hyperplane conjecture Isotropic constant ℓp -Sphere Random polytope Stochastic geometry