Cheeger's isoperimetric problem for Gaussian mixtures

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.