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On Aleksandrov-Fenchel Inequalities for k-Convex Domains

Abstract

In this lecture notes, we will discuss the classical Aleksandrov-Fenchel inequalities for quermassintegrals on convex domains and pose the problem of how to extend the inequalities to non-convex domains. We will survey some recent progress on the problem, then report some of our joint work [9] in which we generalize the k-th stage of the inequalities to a class of (k + 1)-convex domains. Our proof, following the earlier work of Castillion [8] for k =  1 case of the inequalities, uses the method of optimal transport.

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References

  1. 1.

    A.D. Aleksandrov, Zur Theorie der gemischten Volumina von konvexen Körpern, II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen, Mat. Sb. (N.S.) 2 (1937), 1205-1238 (in Russian).

  2. 2.

    A.D. Aleksandrov, Zur Theorie der gemischten Volumina von konvexen Körpern, III. Die Erweiterung zweeier Lehrsatze Minkowskis über die konvexen Polyeder auf beliebige konvexe Flachen, Mat. Sb. (N.S.) 3 (1938), 27-46 (in Russian).

  3. 3.

    Alesker S., Dar S., Milman V. (1999) A remarkable measure preserving diffeomorphism between two convex bodies in \({\mathbb {R}^n}\) . Geom. Dedicata 74: 201–212

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

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

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Caffarelli L.A. (1992) Boundary regularity of maps with convex potentials. Comm. Pure Appl. Math. 45: 1141–1151

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

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

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Caffarelli L.A. (1996) Boundary regularity of maps with convex potentials. II. Ann. Math. 144: 453–496

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Castillon P. (2010) Submanifolds, isoperimetric inequalities and optimal transportation, J. Funct. Anal. 259: 79–103

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    S.-Y. A. Chang and Yi Wang, The Aleksandrov-Fenchel inequalities for Quermassintegrals on k-convex domains, preprint, 2010.

  10. 10.

    S.-Y. A. Chang and Yi Wang, On a class of Michael-Simon inequalities, in preparation, 2011.

  11. 11.

    Cordero-Erausquin D., Nazaret B., Villani C. (2004) A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182: 307–322

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    C. De Lellis; S. Müller, Optimal rigidity estimates for nearly umbilical surfaces, J. Differential Geom. 69 (2005), no. 1, 75110.

    Google Scholar 

  13. 13.

    C. De Lellis; P. Topping, Almost Schur Theorem, Arxiv 1003.3527.

  14. 14.

    L.C. Evans; J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom. 33 (1991), 635C681.

    Google Scholar 

  15. 15.

    Garding L. (1959) An inequality for hyperbolic polynomials. J. Math. Mech. 8: 957–965

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Y. Ge; G. Wang, An almost Schur theorem on 4-dimensional manifolds, to appear on Proc. Amer. Math. Soc.

  17. 17.

    Gerhardt C. (1990) Flow of nonconvex hypersurfaces into spheres. J. Differential Geom. 32: 299–314

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Guan P., Li J. (2009) The quermassintegral inequalities for k-convex starshaped domains. Adv. Math. 221: 1725–1732

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Guan P., Wang G. (2004) Geometric inequalities on locally conformally flat manifolds. Duke Math. J. 124: 177–212

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Huisken G., Ilmanen T. (2001) The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differential Geom. 59: 353–437

    MathSciNet  MATH  Google Scholar 

  21. 21.

    G. Loeper, On the regularity of maps solutions of optimal transportation problems, Acta Math. 202 (2009), no. 2, 241-283.

  22. 22.

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

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    H. Minkowski, Theorie der konvexen Körper, insbesondere Begründung ihres Oberflächenbegriffs. Ges. Abh., Leipzig-Berlin 1911, 2, 131-229.

  24. 24.

    Michael J.H., Simon L.M. (1973) Sobolev and mean-value inequalities on generalized submanifolds of \({\mathbb {R}^n}\) . Comm. Pure Appl. Math. 26: 361–379

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Ma X.N., Trudinger N., Wang X.J. (2005) Regularity of potential functions of the optimal transportation problem. Arch. Rational Mech. Anal. 177: 151–183

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Reilly R. (1973) On the Hessian of a function and the curvatures of its graph. Michigan Math. J. 20: 373–383

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Trudinger N. (1994) Isoperimetric inequalities for quermassintegrals. Ann. Inst. H. Poincare Anal. Non Linaire 11: 411–425

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Urbas J. (1990) On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205: 355–372

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    C. Villani, Topics in optimal transportation, Graduate studies in mathematics, vol. 58. American Mathematical Society, Providence (2003).

  30. 30.

    C. Villani, Optimal transport : old and new, Grundlehren Math. Wiss. 338, Springer, Berlin, 2009.

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Correspondence to Sun-Yung Alice Chang.

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The research of the first author is partially supported by NSF grant DMS-0758601.

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Chang, SY.A., Wang, Y. On Aleksandrov-Fenchel Inequalities for k-Convex Domains. Milan J. Math. 79, 13 (2011). https://doi.org/10.1007/s00032-011-0159-2

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Mathematics Subject Classification (2010)

  • 52A20
  • 53A07
  • 52A39
  • 52A4

Keywords

  • Aleksandrov-Fenchel inequality
  • optimal transport map
  • mixed volume
  • quermassintegral