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The Journal of Geometric Analysis

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

Perimeter under Multiple Steiner Symmetrizations

  • Almut Burchard
  • Gregory R. Chambers
Article
  • 158 Downloads

Abstract

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

Keywords

Hausdorff Distance Nonnegative Continuous Function Dimensional Hausdorff Measure Coordinate Hyperplane Independent Direction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

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    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) CrossRefzbMATHGoogle Scholar
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    Chlebík, M., Cianchi, A., Fusco, N.: The perimeter inequality under Steiner symmetrization: cases of equality. Ann. Math. 162, 525–555 (2005) CrossRefzbMATHGoogle Scholar
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    Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems. Cambridge University Press, Cambridge (2012) CrossRefzbMATHGoogle Scholar
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    Ambrosio, L., Colesanti, A., Villa, E.: Outer Minkowski content for some classes of closed sets. Math. Ann. 342(4), 727–748 (2008) CrossRefzbMATHMathSciNetGoogle 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

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