Total variation bound for Hadwiger's functional using Stein's method

TL;DR

This paper derives explicit bounds on how closely the distribution of the information content of a convex body's associated random vector approximates a Gaussian distribution in high dimensions, using Stein's method.

Contribution

It introduces a novel application of Stein's method combined with Brascamp-Lieb inequality to bound the total variation distance for Hadwiger's functional in high dimensions.

Findings

Explicit total variation bounds are obtained.

The results show Gaussian approximation improves with increasing dimension.

The method applies to convex bodies with specific density functions.

Abstract

Let $K$ be a convex body in $\mathbb{R}^d$. Let $X_K$ be a $d$-dimensional random vector distributed according to the Hadwiger-Wills density $\mu_K$ associated with $K$, defined as $\mu_K(x)=ce^{-\pi {\rm dist}^2(x,K)}$, $x\in \mathbb{R}^d$. Finally, let the information content $H_K$ be defined as $H_K={\rm dist}^2(X_K,K)$. The goal of this paper is to study the fluctuations of $H_K$ around its expectation as the dimension $d$ go to infinity. Relying on Stein's method and Brascamp-Lieb inequality, we compute an explicit bound for the total variation distance between $H_K$ and its Gaussian counterpart.