On the Cheeger sets in strips and non-convex domains

TL;DR

This paper investigates the Cheeger problem in non-convex planar domains, especially strips, providing new estimates for the Cheeger constant and identifying properties shared with convex domains, while highlighting differences in general non-convex cases.

Contribution

The paper offers a stronger estimate for the Cheeger constant of strips and demonstrates that strips share key Cheeger properties with convex domains, unlike general non-convex domains.

Findings

Strips have a Cheeger constant estimate stronger than previous results.

Strips share key Cheeger properties with convex domains.

Counterexamples show these properties do not hold for all non-convex domains.

Abstract

In this paper we consider the Cheeger problem for non-convex domains, with a particular interest in the case of planar strips, which have been extensively studied in recent years. Our main results are an estimate on the Cheeger constant of strips, which is stronger than the previous one known from a recent result by D. Krejcirik and the second-named author, and the proof that strips share with convex domains a number of crucial properties with respect to the Cheeger problem. Moreover, we present several counterexamples showing that the same properties are not valid for generic non-convex domains.