On the isotropic constant of random polytopes with vertices on an ℓp-Sphere

Hörrmann, Julia; Prochno, Joscha; Thäle, Christoph

Abstract

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.

Citation

Hörrmann, J., Prochno, J., & Thäle, C. (2018). On the isotropic constant of random polytopes with vertices on an ℓp-Sphere. The Journal of geometric analysis, 28(1), 405-426. https://doi.org/10.1007/s12220-017-9826-z

• Journal Article Type: Article
• Acceptance Date: May 30, 2016
• Online Publication Date: Mar 7, 2017
• Publication Date: Jan 1, 2018
• Deposit Date: Jun 28, 2018
• Journal: The Journal of Geometric Analysis
• Print ISSN: 1050-6926
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 28
• Issue: 1
• Pages: 405-426
• DOI: https://doi.org/10.1007/s12220-017-9826-z
• Keywords: Asymptotic convex geometry Cone measure Hyperplane conjecture Isotropic constant ℓp -Sphere Random polytope Stochastic geometry
• Public URL: https://hull-repository.worktribe.com/output/898805
• Publisher URL: https://link.springer.com/article/10.1007%2Fs12220-017-9826-z
• Contract Date: Jun 28, 2018