On the moments of the volume for random convex chains

2025/10/17 by Anna Gusakova, Gusakova, Anna, Anna Muranova +1 · 1 citation
Mathematics · #05A15 #33C05 #52A22 #60D05 #Combinatorics (math.CO) #FOS: Mathematics #Geometry and complex manifolds #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2510.15807

openalex publication_date 2025/10/17 · openalex created_date 2025/10/21 · openalex updated_date 2026/07/28

Abstract

Let T be the triangle in the plane with vertices (0, 0), (0,1) and (0, 1). The convex hull T_n of points (0, 1), (1, 0) and n independent random points uniformly distributed in T is the random convex chain. In this paper we study the moments of the volume of random polytope T_n and derive exact formulas for k-th moments for any integer k≥ 0. As an intermediate result, we find an explicit representation for the probability generating function of the number of vertices of T_n, from which an alternative formula for the probability that T_n has k vertices follows.

Citations

• Random convex chains through the lens of analytic combinatorics
• On Random Simplex Picking Beyond the Blashke Problem
• Variance expansion and Berry-Esseen bound for the number of vertices of a random polygon in a polygon
• The Duality of the Volumes and the Numbers of Vertices of Random Polytopes

Cited by

• Random convex chains through the lens of analytic combinatorics

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