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Multiple Sets Exponential Concentration and Higher Order Eigenvalues

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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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Authors and Affiliations

  1. MAP5 (UMR CNRS 8145), Université Paris Descartes, 45 rue des Saints-Pères, 75270, Paris Cedex 6, France

    Nathaël Gozlan

  2. MRU, Université du Luxembourg, 6 Avenue de la Fonte, L-4363, Esch-sur-Alzette, Luxembourg

    Ronan Herry

  3. LAMA (UMR CNRS 8050), Université Paris Est Marne la Vallée, 5 bd Descartes, 77454, Marne la Vallée Cedex 2, France

    Ronan Herry

Authors
  1. Nathaël Gozlan
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  2. Ronan Herry
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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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  • DOI: https://doi.org/10.1007/s11118-018-9743-1

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