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

Hörrmann, Julia; Prochno, Joscha; Thäle, Christoph

Authors

Julia Hörrmann

Joscha Prochno

Christoph Thäle



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.

Journal Article Type Article
Publication Date Jan 1, 2018
Journal The Journal of Geometric Analysis
Print ISSN 1050-6926
Electronic ISSN 1559-002X
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 28
Issue 1
Pages 405-426
APA6 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. doi:10.1007/s12220-017-9826-z
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
Publisher URL https://link.springer.com/article/10.1007%2Fs12220-017-9826-z
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