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Total variation bound for Hadwiger's functional using Stein's method
Valentin Garino, Ivan Nourdin · 2023-04-13 · via math.PR updates on arXiv.org

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 $μ_K$ associated with $K$, defined as $μ_K(x)=ce^{-π{\rm dist}^2(x,K)}$, $x\in \mathbb{R}^d$. 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.

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