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We investigate the properties of the Cheeger sets of rotationally invariant,bounded domains $\Omega \subset \mathbb{R}^n$. For a rotationally invariantCheeger set $C$, the free boundary $\partial C \cap \Omega$ consists of piecesof Delaunay surfaces, which are rotationally invariant surfaces of constantmean curvature. We show that if $\Omega$ is convex, then the free boundary of$C$ consists only of pieces of spheres and nodoids. This result remains validfor nonconvex domains when the generating curve of $C$ is closed, convex, andof class $\mathcal{C}^{1,1}$. Moreover, we provide numerical evidence of thefact that, for general nonconvex domains, pieces of unduloids or cylinders canalso appear in the free boundary of $C$.

On the Cheeger problem for rotationally invariant domains