Skip to main content
SpringerLink
Log in

Minimality of planes in normed spaces

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We prove that a region in a two-dimensional affine subspace of a normed space V has the least 2-dimensional Hausdorff measure among all compact surfaces with the same boundary. Furthermore, the 2-dimensional Hausdorff area density admits a convex extension to Λ2 V. The proof is based on a (probably) new inequality for the Euclidean area of a convex centrally-symmetric polygon.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Almgren F.J. Jr.: Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Annals of Mathematics 87(2), 321–391 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  2. Burago D., Ivanov S.: On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume. Annals of Mathematics 156(2), 891–914 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burago D., Ivanov S.: Gaussian images of surfaces and ellipticity of surface area functionals. Geometric and Functional Analysis 14(3), 469–490 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Busemann H.: Intrinsic area. Annals of Mathematics 48(2), 234–267 (1947)

    Article  MathSciNet  Google Scholar 

  5. Busemann H.: A theorem on convex bodies of the Brunn–Minkowski type. Proceedings of the National Academy of Sciences USA 35, 27–31 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  6. Busemann H.: Geometries in which the planes minimize area. Annali di Matematica Pura ed Applicata 55(4), 171–189 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  7. Busemann H., Ewald G., Shephard G.C.: Convex bodies and convexity on Grassmann cones. I–IV. Mathematische Annalen 151, 1–41 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  8. Busemann H., Straus E.G.: Area and normality. Pacific Journal of Mathematics 10, 35–72 (1960)

    MathSciNet  MATH  Google Scholar 

  9. Gromov M.: Filling Riemannian manifolds. Journal of Differential Geometry 18, 1–147 (1983)

    MathSciNet  MATH  Google Scholar 

  10. Holmes R.D., Thompson A.C.: n-dimensional area and content in Minkowski spaces. Pacific Journal of Mathematics 85, 77–110 (1979)

    MathSciNet  Google Scholar 

  11. Thompson A.C.: Minkowski Geometry. Encyclopedia of Mathematics and its Applications, Vol. 63. Cambridge University Press, Cambridge (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Dmitri Burago

  2. St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia

    Sergei Ivanov

Authors
  1. Dmitri Burago
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergei Ivanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitri Burago.

Additional information

D. Burago was partially supported by NSF grant DMS-0905838. S. Ivanov was partially supported by RFBR grant 11-01-00302-a.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burago, D., Ivanov, S. Minimality of planes in normed spaces. Geom. Funct. Anal. 22, 627–638 (2012). https://doi.org/10.1007/s00039-012-0170-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-012-0170-y

Keywords and phrases

Mathematics Subject Classification (1991)

Search

Navigation