Random Inscribing Polytopes

Ross M Richardson,Lei Wu,Van H Vu

doi:10.46298/dmtcs.3459

Abstract

For convex bodies $K$ with $\mathcal{C}^2$ boundary in $\mathbb{R}^d$, we provide results on the volume of random polytopes with vertices chosen along the boundary of $K$ which we call $\textit{random inscribing polytopes}$. In particular, we prove results concerning the variance and higher moments of the volume, as well as show that the random inscribing polytopes generated by the Poisson process satisfy central limit theorem.

Highlights

Let X be a set in Rd and let x1, . . . , xn be independent random points chosen according to some distribution μ on X
The convex hull of the xi’s is called a random polytope and its study is an active area of research which links together combinatorics, geometry and probability
Let K be a convex body in Rd with volume one and x1, . . . , xn be independent random points chosen according to the uniform distribution on K. We denote this random polytope by Kn. Another one, which we call “inscribing polytopes”, begins with a convex body K, but the points are chosen from the surface of K, with respect to a properly defined measure

Summary

Introduction

Let X be a set in Rd and let x1, . . . , xn be independent random points chosen according to some distribution μ on X. We denote this random polytope by Kn. Another one, which we call “inscribing polytopes”, begins with a convex body K, but the points are chosen from the surface of K, with respect to a properly defined measure (see [14] for the definition of such a measure). The main goal of the theory of random polytopes is to understand the asymptotic behavior of key functionals on Kn, such as the volume or the number of vertices.

Objectives

Results

Conclusion