Comparison of f-vectors of Random Polytopes to the Gaussian Distribution

TL;DR

This paper provides numerical evidence that the components of f-vectors of random polytopes, generated from various distributions, tend to follow a joint Gaussian distribution as the number of points increases.

Contribution

It offers the first numerical validation that f-vector components of diverse random polytopes are approximately Gaussian in large-sample regimes.

Findings

f-vector components are approximately jointly Gaussian for large n

Numerical evidence supports central limit theorem-like behavior across different distributions

Results extend understanding of the probabilistic structure of random polytopes

Abstract

Choose n random, independent points in R^d according to a fixed distribution. The convex hull of these points is a random polytope. In some cases, central limit theorems have been proven for the components of f-vectors of random polytopes constructed in this and similar ways. In this paper, we provide numerical evidence that the components of the f-vectors of random polytopes generated according to five different distributions are approximately jointly Gaussian for large n.