Asymptotic expected $T$-functionals of random polytopes with applications to $L_p$ surface areas

TL;DR

This paper derives an asymptotic formula for the expected $T$-functional of random polytopes formed from i.i.d. points on a sphere, with applications to approximating the sphere via $L_p$ surface areas.

Contribution

It provides a new asymptotic formula for the expected $T$-functional of random polytopes on the sphere, extending understanding of geometric approximation.

Findings

Derived an asymptotic formula for the expected $T$-functional.

Applied results to approximate the sphere using random polytopes.

Connected $T$-functionals to $L_p$ surface area differences.

Abstract

An asymptotic formula is proved for the expected $T$-functional of the convex hull of independent and identically distributed random points sampled from the Euclidean unit sphere in $\mathbb{R}^n$ according to an arbitrary positive continuous density. As an application, the approximation of the sphere by random polytopes in terms of $L_p$ surface area differences is discussed.