  In this paper we consider the generalization of the Cheeger problem which
comes by considering the ratio between the perimeter and a certain power of the
volume. This generalization has been already sometimes treated, but some of the
main properties were still not studied, and our main aim is to fill this gap.
We will show that most of the first important properties of the classical
Cheeger problem are still valid, but others fail; more precisely, long and thin
rectangles will give a counterexample to the property of Cheeger sets of being
the union of all the balls of a certain radius, as well as to the uniqueness.
The shape of Cheeger set for rectangles and strips is then studied as well as
their Cheeger constant.
