Skip to main content
SpringerLink
Log in

Asymmetric \(L_{p}\) convexification and the convex Lorentz–Sobolev inequality

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

An asymmetric version of the concept of \(L_{p}\) convexification of level sets is obtained for piecewise affine functions. Applications to the anisotropic convex Lorentz–Sobolev inequality and the anisotropic Pólya–Szegö principle are discussed.

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. Aleksandrov, A.D.: On the theory of mixed volumes. III. Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies. Mat. Sbornik N.S. 3, 27–46 (1938)

    Google Scholar 

  2. Alvino, A., Ferone, V., Trombetti, G.: Convex symmetrization and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 14(2), 275–293 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)

    MATH  MathSciNet  Google Scholar 

  4. Bandle, C.: Isoperimetric Inequalities and Applications. Pitman, Boston (1980)

    MATH  Google Scholar 

  5. Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  6. Chou, K.-S., Wang, X.-J.: The \(L_{p}\)-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv. Math. 205, 33–83 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cianchi, A., Lutwak, E., Yang, D., Zhang, G.: Affine Moser–Trudinger and Morrey–Sobolev inequalities. Calc. Var. Partial Differ. Equ. 36, 419–436 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  9. Federer, H., Fleming, W.: Normal and integral currents. Ann. Math. 72, 458–520 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fenchel, W., Jessen, B.: Mengenfunktionen und konvexe Körper. Danske Vid. Selskab. Mat.-fys. Medd. 16, 1–31 (1938)

    Google Scholar 

  11. Gardner, R.: The Brunn–Minkowski inequality. Bull. Am. Math. Soc. 39, 355–405 (2002)

    Article  MATH  Google Scholar 

  12. He, B., Liu, S.: The Pólya-Szegö principle and the anisotropic convex Lorentz–Sobolev inequality. Sci. World J. (2014). doi:10.1155/2014/875245

  13. Haberl, C., Schuster, F.: Asymmetric affine \(L_{p}\) Sobolev inequalities. J. Funct. Anal. 257, 641–658 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Haberl, C., Schuster, F.: General \(L_{p}\) affine isoperimetric inequalities. J. Differ. Geom. 83(3), 1–26 (2009)

    MATH  MathSciNet  Google Scholar 

  15. Haberl, C., Schuster, F., Xiao, J.: An asymmetric affine Pólya–Szegö principle. Math. Ann. 352, 517–542 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hug, D., Lutwak, E., Yang, D., Zhang, G.: On the \(L_{p}\) Minkowski problem for polytopes. Discrete Comput. Geom. 33, 699–715 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Leoni, G.: A First Course in Sobolev Spaces. Graduate Studies in Mathematics, vol. 105. American Mathematical Society, Providence (2009)

    Google Scholar 

  18. Ludwig, M.: Minkowski valuations. Trans. Am. Math. Soc. 357(10), 41914213 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Lutwak, E., Yang, D., Zhang, G.: Sharp affine \(L_{p}\) Sobolev inequalities. J. Differ. Geom. 62, 17–38 (2002)

    MATH  MathSciNet  Google Scholar 

  23. Lutwak, E., Yang, D., Zhang, G.: Optimal Sobolev norms and the \(L_{p}\) Minkowski problem. Int. Math. Res. Not. 1, 1–21 (2006)

    MathSciNet  Google Scholar 

  24. Ma, D.: Asymmetric anisotropic fractional Sobolev norms. Arch. Math. (Basel) 103, 167–175 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  25. Maz’ya, V.G.: Classes of domains and imbedding theorems for function spaces. Sov. Math. Dokl. 1, 882–885 (1960)

    MATH  Google Scholar 

  26. Maz’ya, V.G.: Sobolev Spaces. Springer, Berlin (1985)

    Google Scholar 

  27. Minkowski, H.: Allgemeine Lehrsätze über die konvexen Polyeder. Nachr. Ges. Wiss. Göttingen, pp. 198–219 (1897)

  28. Minkowski, H.: Über die Begriffe Länge, Oberfläche und Volumen. Jahresber. Deutsch. Math.-Ver. 9, 115–121 (1901)

    MATH  Google Scholar 

  29. Minkowski, H.: Volumen und Oberfläche. Math. Ann. 57, 447–495 (1903)

    Article  MATH  MathSciNet  Google Scholar 

  30. Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  31. Parapatits, L.: SL(\(n\))-contravariant \(L_{p}\)-Minkowski valuations. Trans. Am. Math. Soc. 366, 1195–1211 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  32. Parapatits, L.: SL(\(n\))-covariant \(L_{p}\)-Minkowski valuations. J. Lond. Math. Soc. 89, 397–414 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  33. Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton (1951)

    MATH  Google Scholar 

  34. Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  35. Schneider, R.: Convex Bodies: the Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  36. Schuster, F.E., Weberndorfer, M.: Volume inequalities for asymmetric Wulff shapes. J. Differ. Geom. 92, 263–283 (2012)

    MATH  MathSciNet  Google Scholar 

  37. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  38. Weberndorfer, M.: Shadow systems of asymmetric \(L_{p}\) zonotopes. Adv. Math. 240, 613–635 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  39. Xiao, J.: The sharp Sobolev and isoperimetric inequalities split twice. Adv. Math. 211, 417–435 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

Download references

Acknowledgments

The author wants to thank Monika Ludwig from the Vienna University of Technology for her valuable advice and both referees for their improvement suggestions leading to this version of the article.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Salzburg, Hellbrunner Strasse 34, 5020, Salzburg, Austria

    Michael Ober

Authors
  1. Michael Ober
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Ober.

Additional information

Communicated by A. Jüngel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ober, M. Asymmetric \(L_{p}\) convexification and the convex Lorentz–Sobolev inequality. Monatsh Math 179, 113–127 (2016). https://doi.org/10.1007/s00605-015-0760-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-015-0760-5

Keywords

Mathematics Subject Classification

Search

Navigation