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 $\gamma :{\mathbb{R}}^n\to (0,\mathrm{\infty })$ 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.