Monatshefte für Mathematik

, Volume 179, Issue 1, pp 113–127 | Cite as

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

Article

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.

Keywords

Asymmetric \(L_{p}\) convexification Asymmetric convex Lorentz–Sobolev inequality Asymmetric Pólya–Szegö principle 

Mathematics Subject Classification

52A40 52A39 

Notes

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.

References

  1. 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. 2.
    Alvino, A., Ferone, V., Trombetti, G.: Convex symmetrization and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 14(2), 275–293 (1997)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)MATHMathSciNetGoogle Scholar
  4. 4.
    Bandle, C.: Isoperimetric Inequalities and Applications. Pitman, Boston (1980)MATHGoogle Scholar
  5. 5.
    Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Springer, Berlin (1988)MATHCrossRefGoogle Scholar
  6. 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)MATHMathSciNetCrossRefGoogle Scholar
  7. 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)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Federer, H., Fleming, W.: Normal and integral currents. Ann. Math. 72, 458–520 (1960)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fenchel, W., Jessen, B.: Mengenfunktionen und konvexe Körper. Danske Vid. Selskab. Mat.-fys. Medd. 16, 1–31 (1938)Google Scholar
  11. 11.
    Gardner, R.: The Brunn–Minkowski inequality. Bull. Am. Math. Soc. 39, 355–405 (2002)MATHCrossRefGoogle Scholar
  12. 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. 13.
    Haberl, C., Schuster, F.: Asymmetric affine \(L_{p}\) Sobolev inequalities. J. Funct. Anal. 257, 641–658 (2009)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Haberl, C., Schuster, F.: General \(L_{p}\) affine isoperimetric inequalities. J. Differ. Geom. 83(3), 1–26 (2009)MATHMathSciNetGoogle Scholar
  15. 15.
    Haberl, C., Schuster, F., Xiao, J.: An asymmetric affine Pólya–Szegö principle. Math. Ann. 352, 517–542 (2012)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Hug, D., Lutwak, E., Yang, D., Zhang, G.: On the \(L_{p}\) Minkowski problem for polytopes. Discrete Comput. Geom. 33, 699–715 (2005)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Leoni, G.: A First Course in Sobolev Spaces. Graduate Studies in Mathematics, vol. 105. American Mathematical Society, Providence (2009)Google Scholar
  18. 18.
    Ludwig, M.: Minkowski valuations. Trans. Am. Math. Soc. 357(10), 41914213 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ludwig, M., Xiao, J., Zhang, G.: Sharp convex Lorentz–Sobolev inequalities. Math. Ann. 350, 169–197 (2011)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Lutwak, E.: The Brunn–Minkowski–Firey theory I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)MATHMathSciNetGoogle Scholar
  21. 21.
    Lutwak, E.: The Brunn–Minkowski–Firey theory II. Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Lutwak, E., Yang, D., Zhang, G.: Sharp affine \(L_{p}\) Sobolev inequalities. J. Differ. Geom. 62, 17–38 (2002)MATHMathSciNetGoogle Scholar
  23. 23.
    Lutwak, E., Yang, D., Zhang, G.: Optimal Sobolev norms and the \(L_{p}\) Minkowski problem. Int. Math. Res. Not. 1, 1–21 (2006)MathSciNetGoogle Scholar
  24. 24.
    Ma, D.: Asymmetric anisotropic fractional Sobolev norms. Arch. Math. (Basel) 103, 167–175 (2014)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Maz’ya, V.G.: Classes of domains and imbedding theorems for function spaces. Sov. Math. Dokl. 1, 882–885 (1960)MATHGoogle Scholar
  26. 26.
    Maz’ya, V.G.: Sobolev Spaces. Springer, Berlin (1985)Google Scholar
  27. 27.
    Minkowski, H.: Allgemeine Lehrsätze über die konvexen Polyeder. Nachr. Ges. Wiss. Göttingen, pp. 198–219 (1897)Google Scholar
  28. 28.
    Minkowski, H.: Über die Begriffe Länge, Oberfläche und Volumen. Jahresber. Deutsch. Math.-Ver. 9, 115–121 (1901)MATHGoogle Scholar
  29. 29.
    Minkowski, H.: Volumen und Oberfläche. Math. Ann. 57, 447–495 (1903)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238 (1978)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Parapatits, L.: SL(\(n\))-contravariant \(L_{p}\)-Minkowski valuations. Trans. Am. Math. Soc. 366, 1195–1211 (2014)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Parapatits, L.: SL(\(n\))-covariant \(L_{p}\)-Minkowski valuations. J. Lond. Math. Soc. 89, 397–414 (2014)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton (1951)MATHGoogle Scholar
  34. 34.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)MATHGoogle Scholar
  35. 35.
    Schneider, R.: Convex Bodies: the Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  36. 36.
    Schuster, F.E., Weberndorfer, M.: Volume inequalities for asymmetric Wulff shapes. J. Differ. Geom. 92, 263–283 (2012)MATHMathSciNetGoogle Scholar
  37. 37.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Weberndorfer, M.: Shadow systems of asymmetric \(L_{p}\) zonotopes. Adv. Math. 240, 613–635 (2013)MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Xiao, J.: The sharp Sobolev and isoperimetric inequalities split twice. Adv. Math. 211, 417–435 (2007)MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Zhang, G.: The affine Sobolev inequality. J. Differ. Geom. 53, 183–202 (1999)MATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SalzburgSalzburgAustria

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