Multiple Sets Exponential Concentration and Higher Order Eigenvalues

On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k -th eigenvalues of the metric Laplacian gives exponential improved concentration with k sets. On compact Riemannian manifolds, this allows us to recover estimates on the eigenvalues of the Laplace-Beltrami operator in the spirit of an inequality of [ 11 ].