Skip to main content

A mass transportation approach to quantitative isoperimetric inequalities

Abstract

A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov’s proof of the isoperimetric inequality and the Brenier-McCann Theorem. A sharp quantitative version of the Brunn-Minkowski inequality for convex sets is proved as a corollary.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Alberti, A., Ambrosio, L.: A geometrical approach to monotone functions in ℝn. Math. Z. 230(2), 259–316 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  2. 2.

    Ambrosio, L., Caselles, V., Masnou, S., Morel, J.-M.: Connected components of sets of finite perimeter and applications to image processing. J. Eur. Math. Soc. (JEMS) 3(1), 39–92 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  3. 3.

    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)

    MATH  Google Scholar 

  4. 4.

    Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11(4), 573–598 (1976)

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Bernstein, F.: Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene. Math. Ann. 60, 117–136 (1905)

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    Bianchi, G., Egnell, H.: A note on the Sobolev inequality. J. Funct. Anal. 100(1), 18–24 (1991)

    MATH  Article  MathSciNet  Google Scholar 

  7. 7.

    Bonnesen, T.: Über die isoperimetrische Defizite ebener Figuren. Math. Ann. 91, 252–268 (1924)

    MATH  Article  MathSciNet  Google Scholar 

  8. 8.

    Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)

    MATH  Article  MathSciNet  Google Scholar 

  9. 9.

    Brothers, J.E., Morgan, F.: The isoperimetric theorem for general integrands. Mich. Math. J. 41(3), 419–431 (1994)

    MATH  Article  MathSciNet  Google Scholar 

  10. 10.

    Caffarelli, L.A.: The regularity of mappings with a convex potential. J. Am. Math. Soc. 5(1), 99–104 (1992)

    MATH  MathSciNet  Google Scholar 

  11. 11.

    Cianchi, A., Fusco, N., Maggi, F., Pratelli, A.: The sharp Sobolev inequality in quantitative form. J. Eur. Math. Soc. (JEMS) 11(5), 1105–1139 (2009)

    MATH  Article  MathSciNet  Google Scholar 

  12. 12.

    Cordero-Erausquin, D., Nazaret, B., Villani, C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182(2), 307–332 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    Dacorogna, B., Pfister, C.-E.: Wulff theorem and best constant in Sobolev inequality. J. Math. Pures Appl. (9) 71(2), 97–118 (1992)

    MathSciNet  Google Scholar 

  14. 14.

    Dinghas, A.: Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen, Z. Kristallogr., Mineral. Petrogr. 105, (1944) (in German)

  15. 15.

    Diskant, V.I.: Stability of the solution of a Minkowski equation. Sib. Mat. Z. 14, 669–673,696 (1973) (in Russian)

    MATH  MathSciNet  Google Scholar 

  16. 16.

    De Giorgi, E.: Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita. Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (8) 5, 33–44 (1958) (in Italian)

    MathSciNet  Google Scholar 

  17. 17.

    Dolzmann, G., Müller, S.: Microstructures with finite surface energy: the two-well problem. Arch. Ration. Mech. Anal. 132(2), 101–141 (1995)

    MATH  Article  Google Scholar 

  18. 18.

    Esposito, L., Fusco, N., Trombetti, C.: A quantitative version of the isoperimetric inequality: the anisotropic case. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4(4), 619–651 (2005)

    MATH  MathSciNet  Google Scholar 

  19. 19.

    Figalli, A., Maggi, F., Pratelli, A.: A refined Brunn-Minkowski inequality for convex sets. Ann. Inst. Henri Poincaré Anal. Non Linèaire 26(6), 2511–2519 (2009)

    MATH  Article  MathSciNet  Google Scholar 

  20. 20.

    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969), xiv+676 pp.

    MATH  Google Scholar 

  21. 21.

    Fonseca, I., Müller, S.: A uniqueness proof for the Wulff theorem. Proc. R. Soc. Edinb. Sect. A 119(1–2), 125–136 (1991)

    MATH  Google Scholar 

  22. 22.

    Fuglede, B.: Stability in the isoperimetric problem for convex or nearly spherical domains in ℝn. Trans. Am. Math. Soc. 314, 619–638 (1989)

    MATH  MathSciNet  Google Scholar 

  23. 23.

    Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168, 941–980 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  24. 24.

    Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative Sobolev inequality for functions of bounded variation. J. Funct. Anal. 244(1), 315–341 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  25. 25.

    Gardner, R.J.: The Brunn-Minkowski inequality. Bull. Am. Math. Soc. (NS) 39(3), 355–405 (2002)

    MATH  Article  Google Scholar 

  26. 26.

    Groemer, H.: On the Brunn-Minkowski theorem. Geom. Dedic. 27(3), 357–371 (1988)

    MATH  Article  MathSciNet  Google Scholar 

  27. 27.

    Groemer, H.: On an inequality of Minkowski for mixed volumes. Geom. Dedic. 33(1), 117–122 (1990)

    MATH  Article  MathSciNet  Google Scholar 

  28. 28.

    Gurtin, M.E.: On a theory of phase transitions with interfacial energy. Arch. Ration. Mech. Anal. 87(3), 187–212 (1985)

    Article  MathSciNet  Google Scholar 

  29. 29.

    Hadwiger, H., Ohmann, D.: Brunn-Minkowskischer Satz und Isoperimetrie. Math. Z. 66, 1–8 (1956)

    MATH  Article  MathSciNet  Google Scholar 

  30. 30.

    Hall, R.R.: A quantitative isoperimetric inequality in n-dimensional space. J. Reine Angew. Math. 428, 161–176 (1992)

    MATH  MathSciNet  Google Scholar 

  31. 31.

    Hall, R.R., Hayman, W.K., Weitsman, A.W.: On asymmetry and capacity. J. Anal. Math. 56, 87–123 (1991)

    MATH  Article  MathSciNet  Google Scholar 

  32. 32.

    Herring, C.: Some theorems on the free energies of crystal surfaces. Phys. Rev. 82, 87–93 (1951)

    MATH  Article  Google Scholar 

  33. 33.

    John, F.: Extremum Problems with Inequalities as Subsidiary Conditions. Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pp. 187–204. Interscience, New York (1948)

    Google Scholar 

  34. 34.

    Knothe, H.: Contributions to the theory of convex bodies. Mich. Math. J. 4, 39–52 (1957)

    MATH  Article  MathSciNet  Google Scholar 

  35. 35.

    Maggi, F.: Some methods for studying stability in isoperimetric type problems. Bull. Am. Math. Soc. 45, 367–408 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  36. 36.

    Maggi, F., Villani, C.: Balls have the worst best Sobolev inequalities. J. Geom. Anal. 15(1), 83–121 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  37. 37.

    Maz’ja, V.G.: Sobolev Spaces. Springer Series in Soviet Mathematics. Springer, Berlin (1985), xix+486 pp. Translated from the Russian by T.O. Shaposhnikova

    MATH  Google Scholar 

  38. 38.

    McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  39. 39.

    McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153–179 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  40. 40.

    Milman, V.D., Schechtman, G.: Asymptotic Theory of Finite-dimensional Normed Spaces. With an appendix by M. Gromov. Lecture Notes in Mathematics, vol. 1200. Springer, Berlin (1986), viii+156 pp.

    MATH  Google Scholar 

  41. 41.

    Ruzsa, I.Z.: The Brunn-Minkowski inequality and nonconvex sets. Geom. Dedic. 67(3), 337–348 (1997) (English summary)

    MATH  Article  MathSciNet  Google Scholar 

  42. 42.

    Schneider, R.: On the general Brunn-Minkowski theorem. Beitr. Algebra Geom. 34(1), 1–8 (1993)

    MATH  Google Scholar 

  43. 43.

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

    MATH  Article  MathSciNet  Google Scholar 

  44. 44.

    Taylor, J.E.: Crystalline variational problems. Bull. Am. Math. Soc. 84(4), 568–588 (1978)

    MATH  Article  Google Scholar 

  45. 45.

    Van Schaftingen, J.: Anisotropic symmetrization. Ann. Inst. Henri Poincaré Anal. Non Lineaire 23(4), 539–565 (2006)

    MATH  Article  Google Scholar 

  46. 46.

    Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003), xvi+370 pp.

    MATH  Google Scholar 

  47. 47.

    Wulff, G.: Zur Frage der Geschwindigkeit des Wachsturms und der Auflösung der Kristallflächen. Z. Kristallogr. 34, 449–530 (1901)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. Figalli.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Figalli, A., Maggi, F. & Pratelli, A. A mass transportation approach to quantitative isoperimetric inequalities. Invent. math. 182, 167–211 (2010). https://doi.org/10.1007/s00222-010-0261-z

Download citation

Keywords

  • Sobolev Inequality
  • Isoperimetric Inequality
  • Asymmetry Index
  • Trace Inequality
  • Coarea Formula