The Cheeger constant of curved strips

TL;DR

This paper investigates the Cheeger constant and Cheeger sets for curved strip domains, establishing exact results for certain cases and bounds for others, supported by numerical analysis.

Contribution

It provides new insights into Cheeger constants for curved strips, including exact values for complete curves and bounds for non-complete curves.

Findings

Cheeger set equals the strip for complete finite curves.

Cheeger constant is the inverse of the strip's half-width for certain domains.

Numerical results for circular sectors support theoretical findings.

Abstract

We study the Cheeger constant and Cheeger set for domains obtained as strip-like neighbourhoods 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 non-complete 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.