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 Chung, Grigor’yan & Yau, Upper bounds for eigenvalues of the discrete and continuous Laplace operators. Adv. Math. 117(2), 165–178 (1996).