  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 Chung, Grigory'an
and Yau [11].
