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In this paper we consider the generalization of the Cheeger problem whichcomes by considering the ratio between the perimeter and a certain power of thevolume. This generalization has been already sometimes treated, but some of themain 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 classicalCheeger problem are still valid, but others fail; more precisely, long and thinrectangles will give a counterexample to the property of Cheeger sets of beingthe 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 astheir Cheeger constant.

On the generalized Cheeger problem and an application to 2d strips