  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.
