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The p-Affine Capacity Redux

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Abstract

Continuing from Xiao (Adv Math 268:906–914, 2015; J Geom Anal 26:947–966, 2016), this note is devoted to the discovery of new geometric properties of the so-called \([1,n)\ni p\)-affine capacity in the Euclidean n-space.

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References

  1. Federer, H., Ziemer, W.P.: The Lebesgue set of a function whose distribution derivatives are \(p\)-th power summable. Indiana Univ. Math. J. 22, 139–158 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  2. Gardner, R.J., Hartenstine, D.: Capacities, surface area, and radial sums. Adv. Math. 221, 601–625 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

  4. Kubota, T.: Uber die konvex-geschlossenen Mannigfaltigkeiten im n-dimensionalen Raume. Sci. Rep. Tojoku Univ. 14, 85–99 (1925)

    MATH  Google Scholar 

  5. Ludwig, M., Xiao, J., Zhang, G.: Sharp convex Lorentz-Sobolev inequalities. Math. Ann. 350, 169–197 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lutwak, E.: The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lutwak, E.: The Brunn-Minkowski-Firey theory II: affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lutwak, E., Yang, D., Zhang, G.: \(L_p\) affine isoperimetric inequalities. J. Differ. Geom. 56, 111–132 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lutwak, E., Yang, D., Zhang, G.: Sharp affine \(L_p\) Sobolev inequalities. J. Differ. Geom. 62, 17–38 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Maz’ya, V.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations, 2nd, revised and augmented edition. Springer, New York (2011)

  11. Petty, C.M.: Isoperimetric problems. In: Proceedings of the Conference on Convexity and Combinatorial Geometry (Univ. Oklahoma, 1971), pp. 26–41. University of Oklahoma, Norman (1972)

  12. Schneider, R.: Convex bodies: the Brunn-Minkowski theory. In: Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1993)

  13. Troyanov, M.: Parabolicity of manifolds. Siberian Adv. Math. 9, 125–150 (1999)

    MathSciNet  MATH  Google Scholar 

  14. Xiao, J.: The sharp Sobolev and isoperimetric inequalities split twice. Adv. Math. 211, 417–435 (2007). Corrigendum. Adv. Math. 268, 906–914 (2015)

  15. Xiao, J.: The \(p\)-affine capacity. J. Geom. Anal. 26, 947–966 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Xiao, J.: A maximum problem of S.-T. Yau for variational \(p\)-capacity. Adv. Geom. (2016) in press

  17. Xiao, J., Zhang, N.: The relative \(p\)-affine capacity. Proc. Am. Math. Soc. 144, 3537–3554 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Xu, X.: Some results on functional capacity and their applications to \(p\)-Laplacian problems involving measure data. Nonlinear Anal. 27, 17–36 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang, G.: The affine Sobolev inequality. J. Differ. Geom. 53, 183–202 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank the referee for the careful review and the valuable comments improving the paper. And, the author is in part supported by NSERC of Canada (FOAPAL # 202979463102000).

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  1. Department of Mathematics and Statistics, Memorial University, St. John’s, NL, A1C5S7, Canada

    Jie Xiao

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  1. Jie Xiao
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Correspondence to Jie Xiao.

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Xiao, J. The p-Affine Capacity Redux. J Geom Anal 27, 2872–2888 (2017). https://doi.org/10.1007/s12220-017-9785-4

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  • DOI: https://doi.org/10.1007/s12220-017-9785-4

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