Extremal random beta polytopes

TL;DR

This paper investigates the asymptotic behavior of extremal intrinsic volumes of random beta polytopes, providing a conjecture and its solution in two dimensions, with broader implications for $U$-$ ext{max}$ statistics.

Contribution

It introduces a conjecture on extremal intrinsic volumes of beta polytopes and solves it explicitly in two dimensions, extending understanding of their asymptotic properties.

Findings

Established a limit relation for $U$-$ ext{max}$ statistics including perimeter and area.

Solved the conjecture for the case in dimension 2.

Extended asymptotic analysis to extremal intrinsic volumes of beta polytopes.

Abstract

The convex hull of several i.i.d. beta distributed random vectors in $\mathbb R^d$ is called the random beta polytope. Recently, the expected values of their intrinsic volumes, number of faces, normal and tangent angles and other quantities have been calculated, explicitly and asymptotically. In this paper, we aim to investigate the asymptotic behavior of the beta polytopes with extremal intrinsic volumes. We suggest a conjecture and solve it in dimension 2. To this end, we obtain some general limit relation for a wide class of $U$-$\max$ statistics whose kernels include the perimeter and the area of the convex hull of the arguments.