Cheeger's isoperimetric problem for Gaussian mixtures

TL;DR

This paper determines the Cheeger constant and sets for Gaussian mixtures in any dimension, confirming a conjecture and analyzing conditions for unique Cheeger sets, advancing understanding of geometric properties of such distributions.

Contribution

It provides a complete characterization of Cheeger sets for Gaussian mixtures and confirms the conjectured solutions, including conditions for uniqueness.

Findings

Cheeger sets are half-spaces perpendicular to the difference of means.

The Cheeger constant is explicitly determined for Gaussian mixtures.

Conditions for the uniqueness of Cheeger sets are identified.

Abstract

In any dimension $n$, we determine the Cheeger constant and the Cheeger sets of the Gaussian mixture $\mu(x) = p\gamma(x-a) + (1-p)\gamma(x-b)$, where $p \in [0,1]$, $a,b \in \mathbb{R}^n$, and $\gamma : \mathbb{R}^n \to (0,\infty)$ denotes a Gaussian. In particular, we characterize the Cheeger sets for $\mu$ in terms of specific half-spaces perpendicular to $a-b$, thereby confirming the conjectured solution to the Cheeger problem for Gaussian mixtures. Finally, we study the regime of parameters $p,a,b$ in which $\mu$ admits a unique Cheeger set.