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Multiple Sets Exponential Concentration and Higher Order Eigenvalues

Abstract

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].

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Correspondence to Nathaël Gozlan.

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Gozlan, N., Herry, R. Multiple Sets Exponential Concentration and Higher Order Eigenvalues. Potential Anal 52, 203–221 (2020). https://doi.org/10.1007/s11118-018-9743-1

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Keywords

  • Concentration of measure phenomenon
  • Eigenvalues of the Laplacian
  • Poincaré inequality

Mathematics Subject Classification (2010)

  • 35P15
  • 60E15
  • 26D10