Abstract

We strengthen, in two different ways, the so called Borell-Brascamp-Lieb inequality in the class of power concave functions. As examples of applications we obtain two quantitative versions of the Brunn-Minkowski inequality and of the Urysohn inequality for torsional rigidity

Author information

Contact details are reproduced from the original publication and may be historical.

Daria Ghilli

Dip. di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy

daria.ghilli@gmail.com

Paolo Salani

Dip. di Matematica, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy

paolo.salani@unifi.it

References

  1. S. Artstein, B. Klartag, V. Milman: The Santalò point of a function, and a functional form of Santalò inequality, Mathematika 51(1–2) (2004) 33–48.https://doi.org/10.1112/S0025579300015497Google Scholar
  2. K. M. Ball, K. J. Böröczky: Stability of the Prékopa-Leindler inequality, Mathematika 56(2) (2010) 339–356.https://doi.org/10.1112/S002557931000063XGoogle Scholar
  3. K. M. Ball, K. J. Böröczky: Stability of some versions of the Prékopa-Leindler inequality, Monatsh. Math. 163(1) (2011) 1–14.https://doi.org/10.1007/s00605-010-0222-zGoogle Scholar
  4. M. Bianchini, P. Salani: Power concavity for solutions of nonlinear elliptic problems in convex domains, in: Geometric Properties for Parabolic and Elliptic PDEs, R. Magnanini et al. (ed.), INdAM Ser. 2, Springer, Milano (2013) 35–48.https://doi.org/10.1007/978-88-470-2841-8_3Google Scholar
  5. C. Bianchini, P. Salani: Concavity properties for elliptic free boundary problems, Nonlinear Anal. 71(10) (2009) 4461–4470.https://doi.org/10.1016/j.na.2009.03.004Google Scholar
  6. C. Borell: Convex set functions in d-space, Period. Math. Hung. 6(2) (1975) 111–136.https://doi.org/10.1007/BF02018814Google Scholar
  7. C. Borell: Capacitary inequalities of the Brunn-Minkowski type, Math. Ann. 263(2) (1983) 179–184.https://doi.org/10.1007/BF01456879Google Scholar
  8. C. Borell: Greenian potentials and concavity, Math. Ann. 272(1) (1985) 155–160.https://doi.org/10.1007/BF01455935Google Scholar
  9. H. J. Brascamp, E. H. Lieb: Some inequalities for Gaussian measures and the long-range order of one-dimensional plasma, in: Functional Integration and its Applications (London, 1974), A. M. Arthurs (ed.), Clarendon Press, Oxford (1975) 1–14.Google Scholar
  10. H. J. Brascamp, E. H. Lieb: On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22(4) (1976) 366–389.https://doi.org/10.1016/0022-1236(76)90004-5Google Scholar
  11. D. Bucur, I. Fragalà: Lower bounds for the Prékopa-Leindler deficit by some distances modulo translations, J. Convex Analysis 21(1) (2014) 289–305.https://doi.org/10.68381/jca21016Google Scholar
  12. D. Bucur, I. Fragalà, J. Lamboley: Optimal convex shapes for concave functionals, ESAIM, Control Optim. Calc. Var. 18(3) (2012) 693–711.https://doi.org/10.1051/cocv/2011167Google Scholar
  13. P. S. Bullen: Handbook of Means and their Inequalities, Mathematics and its Applications 560, Kluwer, Dordrecht (2003).https://doi.org/10.1007/978-94-017-0399-4Google Scholar
  14. M. Christ: Near equality in the two-dimensional Brunn-Minkowski inequality, arXiv:1206.1965v2 [math.CA] (2012).Google Scholar
  15. M. Christ: Near equality in the Brunn-Minkowski inequality, arXiv:1207.5062v1 [math.CA] (2012).Google Scholar
  16. A. Colesanti: Brunn-Minkowski inequalities for variational functionals and related problems, Adv. Math. 194(1) (2005) 105–140.https://doi.org/10.1016/j.aim.2004.06.002Google Scholar
  17. A. Colesanti, P. Cuoghi, P. Salani: Brunn-Minkowski inequalities for two functionals involving the p-Laplace operator, Appl. Anal. 85(1–3) (2006) 45–66.https://doi.org/10.1080/00036810500277819Google Scholar
  18. A. Colesanti, P. Salani: Brunn-Minkowski inequalities for p-capacity of convex bodies, Math. Ann. 327(3) (2003) 459–479.https://doi.org/10.1007/s00208-003-0460-7Google Scholar
  19. D. Cordero-Erasquin, B. Klartag: Interpolations, convexity and geometric inequalities, in: Geometric Aspects of Functional Analysis, B. Klartag et al. (ed.), Lecture Notes in Math. 2050, Springer, Berlin (2012) 151–168.https://doi.org/10.1007/978-3-642-29849-3_9Google Scholar
  20. A. Dinghas: Über eine Klasse superadditiver Mengenfunktionale von Brunn-Minkowski-Lusternikschem Typus, Math. Z. 68 (1957) 111–125.https://doi.org/10.1007/BF01160335Google Scholar
  21. S. Dubuc: Critères de convexité et inégalités intégrales, Ann. Inst. Fourier 27(1) (1977) 135–165.https://doi.org/10.5802/aif.645Google Scholar
  22. R. Eldan, B. Klartag: Dimensionality and the stability of the Brunn-Minkowski inequality, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 13(4) (2014) 975–1007.https://doi.org/10.2422/2036-2145.201111_005Google Scholar
  23. A. Figalli, D. Jerison: Quantitative stability for the Brunn-Minkowski inequality, arXiv:1502.06513v1 [math.MG] (2014).https://doi.org/10.1016/j.aim.2016.12.018Google Scholar
  24. A. Figalli, F. Maggi, A. Pratelli: A refined Brunn-Minkowski inequality for convex sets, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 2511–2519.https://doi.org/10.1016/j.anihpc.2009.07.004Google Scholar
  25. A. Figalli, F. Maggi, A. Pratelli: A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math. 182(1) (2010) 167–211.https://doi.org/10.1007/s00222-010-0261-zGoogle Scholar
  26. R. J. Gardner: The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39(3) (2002) 355–405.https://doi.org/10.1090/S0273-0979-02-00941-2Google Scholar
  27. D. Ghilli, P. Salani: Stability of isoperimetric type inequalities for some Monge-Ampère functionals, Ann. Mat. Pura Appl. (4) 193(3) (2014) 643–661.https://doi.org/10.1007/s10231-012-0295-5Google Scholar
  28. H. Groemer: On the Brunn-Minkowski theorem, Geom. Dedicata 27(3) (1988) 357–371.https://doi.org/10.1007/BF00181500Google Scholar
  29. G. H. Hardy, J. E. Littlewood, G. Polya: Inequalities, 2nd Ed., Cambridge University Press, Cambridge (1952).Google Scholar
  30. D. Hartenstine: Brunn-Minkowski-type inequalities related to the Monge-Ampère equation, Adv. Nonlinear Stud. 9 (2009) 277–294.https://doi.org/10.1515/ans-2009-0204Google Scholar
  31. R. Henstock, A. M. Macbeath: On the measure of sum sets, I. The theorems of Brunn, Minkowski and Lusternik, Proc. Lond. Math. Soc., III. Ser. 3 (1953) 182–194.https://doi.org/10.1112/plms/s3-3.1.182Google Scholar
  32. K. Ishige, P. Salani: Parabolic power concavity and parabolic boundary value problems, Math. Ann. 358(3–4) (2014) 1091–1117.https://doi.org/10.1007/s00208-013-0991-5Google Scholar
  33. B. Kawohl: Geometrical properties of level sets of solutions to elliptic problems, in: Nonlinear Functional Analysis and its Applications. Part 2 (Berkeley, 1983), F. E. Browder (ed.), Proc. Sympos. Pure Math. 45, Amer. Math. Soc., Providence (1986) 25–36.https://doi.org/10.1090/pspum/045.2/843592Google Scholar
  34. A. U. Kennington: Power concavity and boundary value problems, Indiana Univ. Math. J. 34(3) (1985) 687–704.https://doi.org/10.1512/iumj.1985.34.34036Google Scholar
  35. B. Klartag: Marginals of geometric inequalities, in: Geometric Aspects of Functional Analysis, V. D. Milman et al. (ed.), Lecture Notes in Math. 1910, Springer, Berlin (2007) 133–166.https://doi.org/10.1007/978-3-540-72053-9_9Google Scholar
  36. L. G. Makar-Limanov: The solution of the Dirichlet problem for the equation ∆u = −1 in a convex region, Mat. Zametki 9 (1971) 89–92 (in Russian); Math. Notes 9 (1971) 52–53 (in English).https://doi.org/10.1007/BF01405053Google Scholar
  37. L. Leindler: On a certain converse of Hölder’s inequality II, Acta Sci. Math. 33(3–4) (1972) 217–223.Google Scholar
  38. G. Polya, G. Szëgö: Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematical Studies 27, Princeton University Press, Princeton (1951).https://doi.org/10.1515/9781400882663Google Scholar
  39. A. Prékopa: Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. 32 (1971) 301–316.Google Scholar
  40. R. T. Rockafellar: Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, Princeton (1970).Google Scholar
  41. A. Rossi, P. Salani: Stability for Borell-Brascamp-Lieb inequalities, in: Geometric Aspects of Functional Analysis, Israel Seminar (2014–2016), B. Klartag, E. Milman (eds.), Lecture Notes in Mathematics 2169, Springer, Cham (2016) 339–363.https://doi.org/10.1007/978-3-319-45282-1_22Google Scholar
  42. S. Sakaguchi: Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14(3) (1987) 403–421.Google Scholar
  43. P. Salani: A Brunn-Minkowski inequality for the Monge-Ampère eigenvalue, Adv. Math. 194 (2005) 67–86.https://doi.org/10.1016/j.aim.2004.05.011Google Scholar
  44. P. Salani: Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations in R3, Adv. Math. 229(3) (2012) 1924–1948.https://doi.org/10.1016/j.aim.2011.12.009Google Scholar
  45. P. Salani: Combination and mean width rearrangements of solutions of elliptic equations in convex sets, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 32(4) (2015) 763–783.https://doi.org/10.1016/j.anihpc.2014.04.001Google Scholar
  46. A. Segal: Remark on stability of Brunn-Minkowski and isoperimetric inequalities for convex bodies, in: Geometric Aspects of Functional Analysis, B. Klartag et al. (ed.), Lecture Notes in Math. 2050, Springer, Berlin (2012) 381–391.https://doi.org/10.1007/978-3-642-29849-3_24Google Scholar
  47. R. Schneider: Convex bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, Cambridge (1993).https://doi.org/10.1017/CBO9780511526282Google Scholar
  48. T. Strömberg: The operation of infimal convolution, Diss. Math. 352 (1996) 1–58.Google Scholar

Suggested citation

D. Ghilli, P. Salani. “Quantitative Borell-Brascamp-Lieb Inequalities for Power Concave Functions.” Journal of Convex Analysis 24 (2017), No. 3, 857–888. https://doi.org/10.68381/jca24051

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