# Multivariate Normal Approximation for functionals of random polytopes

## TL;DR

This paper establishes central limit theorems for various geometric functionals of random polytopes formed by Poisson point processes in convex bodies, including intrinsic volumes and the f-vector components.

## Contribution

It provides the first multivariate normal approximation results for multiple geometric functionals of random polytopes, including the Will's functional and the oracle estimator.

## Key findings

- Proves CLTs for intrinsic volumes and f-vector components.
- Derives a multivariate CLT for several geometric functionals.
- Establishes asymptotic normality for the volume estimator.

## Abstract

Consider the random polytope, that is given by the convex hull of a Poisson point process on a smooth convex body in $\mathbb{R}^d$. We prove central limit theorems for continuous motion invariant valuations including the Will's functional and the intrinsic volumes of this random polytope. Additionally we derive a central limit theorem for the oracle estimator, that is an unbiased an minimal variance estimator for the volume of a convex set. Finally we obtain a multivariate limit theorem for the intrinsic volumes and the components of the $\mathbf{f}$-vector of the random polytope.