An overview on the Cheeger problem

TL;DR

This paper provides an overview of the Cheeger problem, exploring its connections with eigenvalue estimates, mean curvature equations, and image segmentation, highlighting both classical and recent results in the field.

Contribution

It synthesizes various results related to the Cheeger problem and discusses its links with multiple mathematical problems, including recent collaborative findings.

Findings

Connections between Cheeger problem and eigenvalue estimates

Relation of Cheeger problem to prescribed mean curvature equations

Application of Cheeger problem concepts to image segmentation

Abstract

This is an expository article presenting various results about the Cheeger problem and its connections with other relevant mathematical problems. In section 2 we synthetically describe three mathematical problems (eigenvalue estimates, prescribed mean curvature equation, ROF model for image segmentation) focusing on their link with the Cheeger problem. Then, a selection of classical as well as more recent results obtained in collaboration with Aldo Pratelli will be described in the remaining sections.