Characterization of Cheeger sets for convex subsets of the plane

Bernd Kawohl and Thomas Lachand-Robert

Vol. 225 (2006), No. 1, 103–118

DOI: 10.2140/pjm.2006.225.103

Abstract

Given a planar convex domain Ω, its Cheeger set 𝒞_Ω is defined as the unique minimizer of |∂X|∕|X| among all nonempty open and simply connected subsets X of Ω. We prove an interesting geometric property of 𝒞_Ω, characterize domains Ω which coincide with 𝒞_Ω and provide a constructive algorithm for the determination of 𝒞_Ω.

Keywords

Cheeger constant, Cheeger set, convexity, constructive algorithm, connection-like coycle

Mathematical Subject Classification 2000

Primary: 52A40, 49Q20, 28A75

Milestones

Received: 10 August 2004

Revised: 15 March 2005

Accepted: 24 March 2005

Published: 1 May 2006