Preprint

Inserted: 7 apr 2026
Last Updated: 7 apr 2026

Pages: 22
Year: 2026
Doi: https://doi.org/10.48550/arXiv.2604.03510

Abstract:

$(N, M)$-clusters are partitions of $\mathbb{R}^d$ into $N+M$ regions, where $N$ chambers have prescribed finite measure and $M$ chambers have infinite measure. Locally minimizing clusters are the configurations which minimize the perimeter among all competitors with compact support satisfying the same measure constraints. The characterization of these partitions has been widely studied for the standard (isotropic) perimeter. In the present paper, we investigate the corresponding problem for anisotropic perimeters, considering a general anisotropy. More specifically, we focus on $(1,2)$-clusters and $(1,3)$-clusters in $\mathbb{R}^2$. Our main results provide a geometric characterization of these local minimizers: for regular (smooth, symmetric, and uniformly convex) anisotropies, we prove that a cluster is a local minimizer if and only if, up to translations, it is a standard anisotropic lens cluster in the $(1,2)$-cluster case, or a standard anisotropic triod cluster in the $(1,3)$-cluster case. In addition, using an approximation argument, we extend the minimizing property of these configurations to general anisotropies.

Keywords: anisotropic perimeter, Isoperimetric Clusters