The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 871–882 | Cite as

Perimeter under Multiple Steiner Symmetrizations

Article
  • 144 Downloads

Abstract

Steiner symmetrization in n linearly independent directions transforms every compact subset of \(\mathbb {R}^{n}\) into a set of finite perimeter.

Notes

Acknowledgements

This work was partially supported by an NSERC Discovery Grant (Burchard) and an NSERC Alexander Graham Bell Canada Graduate Scholarship (Chambers). We would also like to thank Luigi Ambrosio for the proof of Lemma 5.

References

  1. 1.
    Steiner, J.: Einfacher Beweis der isoperimetrischen Hauptsätze. J. Reine Angew. Math. 18, 281–296 (1838) and Gesammelte Werke, Vol. 2, pp. 77–91, G. Reimer, Berlin 1882 (in German) CrossRefMATHGoogle Scholar
  2. 2.
    Chlebík, M., Cianchi, A., Fusco, N.: The perimeter inequality under Steiner symmetrization: cases of equality. Ann. Math. 162, 525–555 (2005) CrossRefMATHGoogle Scholar
  3. 3.
    Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems. Cambridge University Press, Cambridge (2012) CrossRefMATHGoogle Scholar
  4. 4.
    Ambrosio, L., Colesanti, A., Villa, E.: Outer Minkowski content for some classes of closed sets. Math. Ann. 342(4), 727–748 (2008) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Burchard, A., Fortier, M.: Random polarizations. Adv. Math. 334, 550–573 (2013) CrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations