On the isotropic constant of random polytopes with vertices on an ℓ_p-sphere

2016/05/30 by Julia Hörrmann, Joscha Prochno, Hörrmann, Julia +3
Computer Science · Mathematics · #52A20 #52B11 #60D05 #Computational Geometry and Mesh Generation #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1605.09160

openalex publication_date 2016/05/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the ℓ_p-unit sphere of \mathbb 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 D. Alonso-Gutiérrez. The proof requires several different tools including a probabilistic representation of 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. Kwapień.

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