Monotonicity of expected volumes of random polytopes

D. N. Zaporozhets^ab, T. D. Moseeva<sup>a

a</sup> St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
^b Saint Petersburg State University

Abstract: We consider a random polytope in $\mathbb{R}^d$ whose vertices are distributed according to a beta distribution. It is shown that, as the parameters of the beta distribution increase, the expected volume of the polytope decreases. If the number of vertices does not exceed $d+1$ (the simplex case), the same monotonicity holds for all positive integer moments of the volume. The results are extended to a broader class of distributions satisfying a stochastic domination condition for the radial components.

Key words and phrases: beta-polytopes, expected volume, beta distribution, random simplices, convex hull, stochastic dominance, intrinsic volumes, geometric probability, stochastic geometry.

UDC: 519.2

Received: 17.11.2025