Skip to main content
SpringerLink
Log in

Hölder continuity for support measures of convex bodies

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

The support measures of a convex body are a common generalization of the curvature measures and the area measures. With respect to the Hausdorff metric on the space of convex bodies, they are weakly continuous. We provide a quantitative improvement of this result by establishing a Hölder estimate for the support measures in terms of the bounded Lipschitz metric which metrizes the weak convergence. Specializing the result to area measures yields a reverse counterpart to earlier stability estimates, concerning Minkowski’s existence theorem for convex bodies with given area measure.

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. Chazal F., Cohen-Steiner D., Mérigot Q.: Boundary measures for geometric inference. Found. Comput. Math. 10, 221–240 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. V. Diskant, Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference (in Russian), Sibirskii Mat. Zh. 13 (1972), 767–772. English translation: Siberian Math. J. 13 (1973), 529–532.

  3. R. M. Dudley, Real Analysis and Probability. Cambridge University Press, New York, 2002.

  4. H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491.

  5. H. Federer, Geometric Measure Theory, Springer, Berlin, 1969.

  6. P. Goodey, M. Kiderlen, and W. Weil, Spherical projections and liftings in geometric tomography, Adv. Geom. 11 (2011), 1–47.

  7. D. Hug and R. Schneider, Stability results involving surface area measures of convex bodies, Rend. Circ. Mat. Palermo (2) Suppl. 70 (2002), vol. II, 21–51.

  8. R. Schneider, Convex Bodies: The Brunn–Minkowski Theory. 2nd. ed., Cambridge University Press, Cambridge, 2014.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Karlsruhe Institute of Technology, 76128, Karlsruhe, Germany

    Daniel Hug

  2. Mathematisches Institut, Albert-Ludwigs-Universität, 79104, Freiburg, Germany

    Rolf Schneider

Authors
  1. Daniel Hug
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rolf Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Hug.

Additional information

The authors acknowledge support by the German research foundation (DFG) through the research group ‘Geometry and Physics of Spatial Random Systems’ under Grant HU1874/2-1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hug, D., Schneider, R. Hölder continuity for support measures of convex bodies. Arch. Math. 104, 83–92 (2015). https://doi.org/10.1007/s00013-014-0719-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-014-0719-0

Mathematics Subject Classification

Keywords

Search

Navigation