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

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.