A note on the stability of the Cheeger constant of $N$-gons

TL;DR

This paper investigates the stability of the minimal Cheeger constant for regular polygons, providing quantitative inequalities and demonstrating the sharpness of these stability results in terms of set distances.

Contribution

It introduces stability inequalities for the Cheeger constant of polygons, relating minimality to geometric distances, and proves the sharpness of these inequalities.

Findings

Stability inequalities in terms of $L^1$ and Hausdorff distances.

Results are sharp in decay rate and distance notion.

Extends understanding of Cheeger constant stability for polygons.

Abstract

The regular $N$-gon provides the minimal Cheeger constant in the class of all $N$-gons with fixed volume. This result is due to a work of Bucur and Fragal\`a in 2014. In this note, we address the stability of their result in terms of the $L^1$ distance between sets. Furthermore, we provide a stability inequality in terms of the Hausdorff distance between the boundaries of sets in the class of polygons having uniformly bounded diameter. Finally, we show that our results are sharp, both in the exponent of decay and in the notion of distance between sets.