We develop the notion of higher Cheeger constants for a measurable set $Omega subset mathbb{R}^N$. By the $k$-th Cheeger constant we mean the value [h_k(Omega) = inf max {h_1(E_1), dots, h_1(E_k)},] where the infimum is taken over all $k$-tuples of mutually disjoint subsets of $Omega$, and $h_1(E_i)$ is the classical Cheeger constant of $E_i$. We prove the existence of minimizers satisfying additional adjustment conditions and study their properties. A relation between $h_k(Omega)$ and spectral minimal $k$-partitions of $Omega$ associated with the first eigenvalues of the $p$-Laplacian under homogeneous Dirichlet boundary conditions is stated. The results are applied to determine the second Cheeger constant of some planar domains.
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