The Cheeger constant of curved strips

David Krejčiřík and Aldo Pratelli

Vol. 254 (2011), No. 2, 309–333

DOI: 10.2140/pjm.2011.254.309

Abstract

We study the Cheeger constant and Cheeger set for domains obtained as strip-like neighborhoods of curves in the plane. If the reference curve is complete and finite (a “curved annulus”), then the strip itself is a Cheeger set and the Cheeger constant equals the inverse of the half-width of the strip. The latter holds true for unbounded strips as well, but there is no Cheeger set. Finally, for strips about noncomplete finite curves, we derive lower and upper bounds to the Cheeger set, which become sharp for infinite curves. The paper is concluded by numerical results for circular sectors.

Keywords

Cheeger sets, Cheeger constant, curved strips

Mathematical Subject Classification 2010

Primary: 28A75, 49Q20, 35P15, 51M16

Milestones

Received: 15 November 2010

Revised: 2 October 2011

Accepted: 17 October 2011

Published: 27 February 2012