On the probability that convex hull of random points contains the origin

2023/04/25 by Konstantin Tikhomirov, Tikhomirov, Konstantin
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2304.13133

openalex publication_date 2023/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The classical theorem of Wendel provides an exact formula for the probability that the convex hull of independent symmetrically distributed vectors in \mathbb R^d contains the origin as long as the distributions of the vectors are continuous. In this note, we provide an extension to Wendel's theorem for independent random vectors X₁,…,X_n with i.i.d components having a (possibly discrete) symmetric distribution of unit variance. As a related observation, we give sharp estimates on the probability that a random linear program of the form ``max⟨ x,\mathfrak c⟩\quadsubject to ⟨ X_i,x⟩≤ 1, i≤ n'', is bounded.

Related

• Random vectors in the isotropic position
• Convex hulls of random walks, hyperplane arrangements, and Weyl chambers
• About the isotropy constant of random convex sets
• On the geometry of random polytopes
• On the deterministic interior body of random polytopes