The Cheeger constant of curved tubes

  • David KrejčiříkEmail author
  • Gian Paolo Leonardi
  • Petr Vlachopulos


We compute the Cheeger constant of tubular neighbourhoods of complete curves in an arbitrary dimensional Euclidean space and raise a question about curved spherical shells.


Cheeger constant Cheeger set Curved tubes Tubular neighbourhoods of curves Spherical shells 

Mathematics Subject Classification

28A75 35P15 49Q10 49Q15 49Q20 51M16 


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We are grateful to Vladimir Bobkov for pointing out to us the references [4,5]. The research of D.K. was partially supported by the GACR Grant No. 18-08835S and by FCT (Portugal) through project PTDC/MATCAL/4334/2014.


  1. 1.
    Anoop, T.V., Bobkov, V., Sasi, S.: On the strict monotonicity of the first eigenvalue of the \(p\)-Laplacian on annuli. Trans. Am. Math. Soc. 370, 7181–7199 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bellettini, G., Caselles, V., Novaga, M.: The total variation flow in \(\mathbb{R}^N\). J. Differ. Equ. 184, 475–525 (2002)CrossRefGoogle Scholar
  3. 3.
    Bishop, R.L.: There is more than one way to frame a curve. Am. Math. Monthly 82, 246–251 (1975)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bueno, H., Ercole, G.: Solutions of the Cheeger problem via torsion functions. J. Math. Anal. Appl. 381, 263–279 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Demengel, F.: Functions locally almost 1-harmonic. Appl. Anal. 83, 865–896 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gray, A.: Tubes. Addison-Wesley, New York (1990)zbMATHGoogle Scholar
  7. 7.
    Grieser, D.: The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem. Arch. Math. 87, 75–85 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kawohl, B.: Two dimensions are easier. Arch. Math. 107, 423–428 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kawohl, B., Fridman, V.: Isoperimetric estimates for the first eigenvalue of the \(p\)-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carol. 44, 659–667 (2003)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kawohl, B., Lachand-Robert, T.: Characterization of Cheeger sets for convex subsets of the plane. Pac. J. Math. 225, 103–118 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kawohl, B., Schuricht, F.: First eigenfunctions of the \(1\)-Laplacian are viscosity solutions. Commun. Pure Appl. Anal. 14, 329–339 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Krejčiřík, D., Pratelli, A.: The Cheeger constant of curved strips. Pac. J. Math. 254, 309–333 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Leonardi, G.P.: An Overview on the Cheeger Problem. New Trends in Shape Optimization. International Series of Numerical Mathematics, vol. 166, pp. 117–139. Springer, New York (2015)CrossRefGoogle Scholar
  14. 14.
    Leonardi, G.P., Pratelli, A.: On the Cheeger sets in strips and non-convex domains. Calc. Var. Partial Differ. Equ. 55(15), 1–28 (2016)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Leonardi, G.P., Neumayer, R., Saracco, G.: The Cheeger constant of a Jordan domain without necks. Calc. Var. Partial Differ. Equ. 56(164), 1–29 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Leonardi, G.P., Saracco, G.: Two examples of minimal Cheeger sets in the plane. Ann. Mat. Pura Appl. 197(5), 1511–1531 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  18. 18.
    Parini, E.: An introduction to the Cheeger problem. Surv. Math. Appl. 6, 9–22 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • David Krejčiřík
    • 1
    Email author
  • Gian Paolo Leonardi
    • 2
  • Petr Vlachopulos
    • 1
  1. 1.Department of Mathematics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePrague 2Czechia
  2. 2.Department of MathematicsUniversity of TrentoPovo - TrentoItaly

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