Monatshefte für Mathematik

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

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

  • Michael Ober


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.


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

Mathematics Subject Classification

52A40 52A39 



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.


  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)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bandle, C.: Isoperimetric Inequalities and Applications. Pitman, Boston (1980)zbMATHGoogle Scholar
  5. 5.
    Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Springer, Berlin (1988)zbMATHCrossRefGoogle 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)zbMATHMathSciNetCrossRefGoogle 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)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Federer, H., Fleming, W.: Normal and integral currents. Ann. Math. 72, 458–520 (1960)zbMATHMathSciNetCrossRefGoogle 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)zbMATHCrossRefGoogle 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)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Haberl, C., Schuster, F.: General \(L_{p}\) affine isoperimetric inequalities. J. Differ. Geom. 83(3), 1–26 (2009)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Haberl, C., Schuster, F., Xiao, J.: An asymmetric affine Pólya–Szegö principle. Math. Ann. 352, 517–542 (2012)zbMATHMathSciNetCrossRefGoogle 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)zbMATHMathSciNetCrossRefGoogle 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)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Lutwak, E.: The Brunn–Minkowski–Firey theory I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Lutwak, E.: The Brunn–Minkowski–Firey theory II. Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Lutwak, E., Yang, D., Zhang, G.: Sharp affine \(L_{p}\) Sobolev inequalities. J. Differ. Geom. 62, 17–38 (2002)zbMATHMathSciNetGoogle 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)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Maz’ya, V.G.: Classes of domains and imbedding theorems for function spaces. Sov. Math. Dokl. 1, 882–885 (1960)zbMATHGoogle 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)zbMATHGoogle Scholar
  29. 29.
    Minkowski, H.: Volumen und Oberfläche. Math. Ann. 57, 447–495 (1903)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Parapatits, L.: SL(\(n\))-contravariant \(L_{p}\)-Minkowski valuations. Trans. Am. Math. Soc. 366, 1195–1211 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Parapatits, L.: SL(\(n\))-covariant \(L_{p}\)-Minkowski valuations. J. Lond. Math. Soc. 89, 397–414 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton (1951)zbMATHGoogle Scholar
  34. 34.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)zbMATHGoogle 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)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Weberndorfer, M.: Shadow systems of asymmetric \(L_{p}\) zonotopes. Adv. Math. 240, 613–635 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Xiao, J.: The sharp Sobolev and isoperimetric inequalities split twice. Adv. Math. 211, 417–435 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Zhang, G.: The affine Sobolev inequality. J. Differ. Geom. 53, 183–202 (1999)zbMATHGoogle Scholar

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© Springer-Verlag Wien 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SalzburgSalzburgAustria

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