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Capacity and Covering Numbers

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Abstract

We establish the inequality \(1/C_K(E)\ge \int_0^\infty |dK(t)|/N_E(t)\), where E is a compact metric space, K is a kernel function, C K is the associated capacity, and N E (t) denotes the minimal number of sets of diameter t needed to cover E. We give applications to the capacity of generalized Cantor sets, and to the capacity of δ-neighborhoods of a set. We also investigate possible converses to the inequality.

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Authors and Affiliations

  1. Département de mathématiques et de statistique, Université Laval, Québec (QC), Canada, G1V 0A6

    Thomas Ransford & Alexis Selezneff

Authors
  1. Thomas Ransford
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  2. Alexis Selezneff
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Corresponding author

Correspondence to Thomas Ransford.

Additional information

Thomas Ransford is partially supported by grants from NSERC (Canada), FQRNT (Québec) and the Canada Research Chairs program.

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Ransford, T., Selezneff, A. Capacity and Covering Numbers. Potential Anal 36, 223–233 (2012). https://doi.org/10.1007/s11118-011-9226-0

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  • DOI: https://doi.org/10.1007/s11118-011-9226-0

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