Cheeger's isoperimetric problem for Gaussian mixtures
Abstract
In any dimension $n$, we determine the Cheeger constant and the Cheeger sets of the Gaussian mixture $μ(x) = pγ(x-a) + (1-p)γ(x-b)$, where $p \in [0,1]$, $a,b \in \mathbb{R}^n$, and $γ: \mathbb{R}^n \to (0,\infty)$ denotes a Gaussian. In particular, we characterize the Cheeger sets for $μ$ 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 $μ$ admits a unique Cheeger set.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Point processes and geometric inequalities
Nonlinear Partial Differential Equations
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 16 February 2026
- Language
- EN
- Document type
- Preprint