On the generalized Cheeger problem and an application to 2d strips

TL;DR

This paper explores a generalized Cheeger problem involving a ratio of perimeter to a power of volume, analyzing its properties, limitations, and specific shapes like rectangles and strips.

Contribution

It extends the classical Cheeger problem by studying its generalized form, revealing which properties hold and which do not, and analyzing Cheeger sets in 2D strips.

Findings

Most classical properties remain valid in the generalized setting.

Long, thin rectangles serve as counterexamples to certain properties.

The shape and Cheeger constant of rectangles and strips are characterized.

Abstract

In this paper we consider the generalization of the Cheeger problem which comes by considering the ratio between the perimeter and a certain power of the volume. This generalization has been already sometimes treated, but some of the main properties were still not studied, and our main aim is to fill this gap. We will show that most of the first important properties of the classical Cheeger problem are still valid, but others fail; more precisely, long and thin rectangles will give a counterexample to the property of Cheeger sets of being the union of all the balls of a certain radius, as well as to the uniqueness. The shape of Cheeger set for rectangles and strips is then studied as well as their Cheeger constant.