A mass transportation approach to quantitative isoperimetric inequalities

Inventiones Mathematicae

Volumn 182, Issue 1, 2010, Pages 167-211

  Figalli, A ^a   Maggi, F ^b   Pratelli, A ^c  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b UNIVERSITY OF FLORENCE   (Italy)

^c UNIVERSITY OF PAVIA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77955309823     PISSN: 00209910     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00222-010-0261-z     Document Type: Article

References (47)

1. n. Math. Z. 230(2), 259-316 (1999).
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).
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).
4. Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11(4), 573-598 (1976).
5. Bernstein, F.: Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene. Math. Ann. 60, 117-136 (1905).
6. Bianchi, G., Egnell, H.: A note on the Sobolev inequality. J. Funct. Anal. 100(1), 18-24 (1991).
7. Bonnesen, T.: Über die isoperimetrische Defizite ebener Figuren. Math. Ann. 91, 252-268 (1924).
8. Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375-417 (1991).
9. Brothers, J. E., Morgan, F.: The isoperimetric theorem for general integrands. Mich. Math. J. 41(3), 419-431 (1994).
10. Caffarelli, L. A.: The regularity of mappings with a convex potential. J. Am. Math. Soc. 5(1), 99-104 (1992).
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).
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).
13. Dacorogna, B., Pfister, C.-E.: Wulff theorem and best constant in Sobolev inequality. J. Math. Pures Appl. (9) 71(2), 97-118 (1992).
14. Dinghas, A.: Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen, Z. Kristallogr., Mineral. Petrogr. 105, (1944) (in German).
15. Diskant, V. I.: Stability of the solution of a Minkowski equation. Sib. Mat. Z. 14, 669-673, 696 (1973) (in Russian).
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).
17. Dolzmann, G., Müller, S.: Microstructures with finite surface energy: the two-well problem. Arch. Ration. Mech. Anal. 132(2), 101-141 (1995).
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).
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).
20. Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969), xiv+676 pp.
21. Fonseca, I., Müller, S.: A uniqueness proof for the Wulff theorem. Proc. R. Soc. Edinb. Sect. A 119(1-2), 125-136 (1991).
22. n. Trans. Am. Math. Soc. 314, 619-638 (1989).
23. Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168, 941-980 (2008).
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).
25. Gardner, R. J.: The Brunn-Minkowski inequality. Bull. Am. Math. Soc. (NS) 39(3), 355-405 (2002).
26. Groemer, H.: On the Brunn-Minkowski theorem. Geom. Dedic. 27(3), 357-371 (1988).
27. Groemer, H.: On an inequality of Minkowski for mixed volumes. Geom. Dedic. 33(1), 117-122 (1990).
28. Gurtin, M. E.: On a theory of phase transitions with interfacial energy. Arch. Ration. Mech. Anal. 87(3), 187-212 (1985).
29. Hadwiger, H., Ohmann, D.: Brunn-Minkowskischer Satz und Isoperimetrie. Math. Z. 66, 1-8 (1956).
30. Hall, R. R.: A quantitative isoperimetric inequality in n-dimensional space. J. Reine Angew. Math. 428, 161-176 (1992).
31. Hall, R. R., Hayman, W. K., Weitsman, A. W.: On asymmetry and capacity. J. Anal. Math. 56, 87-123 (1991).
32. Herring, C.: Some theorems on the free energies of crystal surfaces. Phys. Rev. 82, 87-93 (1951).
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).
34. Knothe, H.: Contributions to the theory of convex bodies. Mich. Math. J. 4, 39-52 (1957).
35. Maggi, F.: Some methods for studying stability in isoperimetric type problems. Bull. Am. Math. Soc. 45, 367-408 (2008).
36. Maggi, F., Villani, C.: Balls have the worst best Sobolev inequalities. J. Geom. Anal. 15(1), 83-121 (2005).
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.
38. McCann, R. J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309-323 (1995).
39. McCann, R. J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153-179 (1997).
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.
41. Ruzsa, I. Z.: The Brunn-Minkowski inequality and nonconvex sets. Geom. Dedic. 67(3), 337-348 (1997) (English summary).
42. Schneider, R.: On the general Brunn-Minkowski theorem. Beitr. Algebra Geom. 34(1), 1-8 (1993).
43. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353-372 (1976).
44. Taylor, J. E.: Crystalline variational problems. Bull. Am. Math. Soc. 84(4), 568-588 (1978).
45. van Schaftingen, J.: Anisotropic symmetrization. Ann. Inst. Henri Poincaré Anal. Non Lineaire 23(4), 539-565 (2006).
46. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003), xvi+370 pp.
47. Wulff, G.: Zur Frage der Geschwindigkeit des Wachsturms und der Auflösung der Kristallflächen. Z. Kristallogr. 34, 449-530 (1901).