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The Cheeger constant of a Jordan domain without necks

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Abstract

We show that the maximal Cheeger set of a Jordan domain \(\Omega \) without necks is the union of all balls of radius \(r = h(\Omega )^{-1}\) contained in \(\Omega \). Here, \(h(\Omega )\) denotes the Cheeger constant of \(\Omega \), that is, the infimum of the ratio of perimeter over area among subsets of \(\Omega \), and a Cheeger set is a set attaining the infimum. The radius r is shown to be the unique number such that the area of the inner parallel set \(\Omega ^r\) is equal to \(\pi r^2\). The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake.

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Notes

  1. Under certain regularity assumptions on A, the outer Minkowski content of A is equal to the perimeter of A; see [3] for a treatment of the subject. In certain cases arising in our setting, these two quantities fail to coincide, but this is of no importance in our application.

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Authors and Affiliations

  1. Dipartimento di Science Fisiche, Informatiche e Matematiche, Università degli Studi di Modena e Reggio Emilia, via Campi 213/b, 41125, Modena, Italy

    Gian Paolo Leonardi

  2. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208, USA

    Robin Neumayer

  3. Department Mathematik, Universität Erlangen-Nürnberg, Cauerst. 11, 91058, Erlangen, Germany

    Giorgio Saracco

Authors
  1. Gian Paolo Leonardi
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  2. Robin Neumayer
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  3. Giorgio Saracco
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Correspondence to Gian Paolo Leonardi.

Additional information

Communicated by L. Ambrosio.

G.P. Leonardi and G. Saracco have been supported by GNAMPA projects: Problemi isoperimetrici e teoria geometrica della misura in spazi metrici (2015) and Variational problems and geometric measure theory in metric spaces (2016). R. Neumayer is supported by the NSF Graduate Research Fellowship under Grant DGE-1110007.

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Leonardi, G.P., Neumayer, R. & Saracco, G. The Cheeger constant of a Jordan domain without necks. Calc. Var. 56, 164 (2017). https://doi.org/10.1007/s00526-017-1263-0

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