Beyond Cheeger's constant

The Cheeger constant of an open set of the Euclidean space is defined by minimizing the ratio “perimeter over volume”, among all its smooth compactly contained subsets. We consider a natural variant of this problem, where the volume of admissible sets is raised to any positive power. We show that for {\it sublinear} powers, all these generalized Cheeger constants are equivalent to the standard one, by means of a universal two-sided estimate. We also show that this equivalence breaks down for {\it superlinear} powers. In this case, some weird phenomena appear. We finally consider the case of convex planar sets and prove an existence result, under optimal assumptions.