On mean outer radii of random polytopes

2012/11/10 by David Alonso–Gutiérrez, Nikos Dafnis, Alonso-Gutierrez, David +5
Computer Science · Mathematics · #05D40 #Computational Geometry and Mesh Generation #FOS: Mathematics #Functional Analysis (math.FA) #Point processes and geometric inequalities #Primary 52A22 #Secondary 52A23 #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1211.2336

openalex publication_date 2012/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

In this paper we introduce a new sequence of quantities for random polytopes. Let K_N=\conv\X₁,...,X_N\ be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body K of \Rⁿ. We prove that the so-called k-th mean outer radius \widetilde R_k(K_N) has order max\√(k),√(log N)\L_K with high probability if n²≤ N≤ e^√(n). We also show that this is also the right order of the expected value of \widetilde R_k(K_N) in the full range n≤ N≤ e^√(n).

Citations

• On the Gaussian behavior of marginals and the mean width of random\n polytopes

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