Advertisement

Potential Analysis

, Volume 22, Issue 3, pp 289–304 | Cite as

The Brunn–Minkowski Inequality for the n-dimensional Logarithmic Capacity of Convex Bodies

  • Andrea Colesanti
  • Paola Cuoghi
Article

Abstract

We define the n-dimensional logarithmic capacity for convex bodies in Rn, with n≥2; then, for this quantity, we prove a Brunn–Minkowski type inequality, and we characterize the corresponding equality case.

Keywords

Brunn–Minkowski inequality logarithmic capacity convex body 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Borell, C.: ‘Capacitary inequalities of the Brunn–Minkowski type’, Math. Ann. 263 (1984), 179–184. Google Scholar
  2. 2.
    Borell, C.: ‘Hitting probability of killed Brownian motion: A study on geometric regularity’, Ann. Sci. Ecole Norm. Supér. Paris 17 (1984), 451–467. Google Scholar
  3. 3.
    Borell, C.: ‘Greenian potentials and concavity’, Math. Ann. 272 (1985), 155–160. Google Scholar
  4. 4.
    Brascamp, H.J. and Lieb, E.H.: ‘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 (1976), 366–389. CrossRefMATHGoogle Scholar
  5. 5.
    Caffarelli, L.A., Jerison, D. and Lieb, E.H.: ‘On the case of equality in the Brunn–Minkowski inequality for capacity’, Adv. Math. 117(2) (1996), 193–207. Google Scholar
  6. 6.
    Caffarelli, L.A. and Spruck, J.: ‘Convexity of solutions to some classical variational problems’, Comm. in P.D.E. 7 (1982), 1337–1379. Google Scholar
  7. 7.
    Colesanti, A.: ‘The Brunn–Minkowski inequality for variational functionals and related problems’, Preprint, 2003. Google Scholar
  8. 8.
    Colesanti, A. and Salani, P.: ‘The Brunn–Minkowski inequality for p-capacity of convex bodies’, Math. Ann. 327 (2003), 459–479. Google Scholar
  9. 9.
    Fekete, M.: ‘Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten’, Math. Z. 17 (1923), 228–249. Google Scholar
  10. 10.
    Gardner, R.J.: ‘The Brunn–Minkowski inequality’, Bull. Amer. Math. Soc. 39(3) (2002), 355–405. Google Scholar
  11. 11.
    Gehring, F.W.: ‘Rings and quasiconformal mappings in space’, Trans. Amer. Math. Soc. 103 (1962), 353–393. MATHGoogle Scholar
  12. 12.
    Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable, Amer. Math. Soc., Providence, RI, 1969. Google Scholar
  13. 13.
    Kołodziej, S.: ‘The logarithmic capacity in Cn’, Ann. Polon. Math. 48(3) (1988), 253–267. Google Scholar
  14. 14.
    Kichenassamy, S. and Veron, L.: ‘Singular solutions of the p-Laplace equations’, Math. Ann. 275 (1986), 599–615. Google Scholar
  15. 15.
    Korevaar, N.: ‘Convexity of level sets for solutions to elliptic ring problems’, Comm. Partial Differential Equations 15 (1990), 541–556. Google Scholar
  16. 16.
    Landkof, N.S.: Foundations of Modern Potential Theory, Springer, Berlin, 1972. Google Scholar
  17. 17.
    Lewis, J.: ‘Capacitary functions in convex rings’, Arch. Rational Mech. Anal. 66 (1977), 201–224. Google Scholar
  18. 18.
    Longinetti, M.: ‘Sulla convessità delle linee di livello di funzioni armoniche’, Boll. Un. Mat. Ital. 2-A(6) (1983), 71–75. Google Scholar
  19. 19.
    Pólya, G. and Szegö, G.: Isoperimetric Inequalities in Mathematical Physics, Princeton Univ. Press, Princeton, 1951. Google Scholar
  20. 20.
    Salani, P.: ‘The Brunn–Minkowski inequality for the Monge–Ampère eigenvalue’, Preprint, 2003. Google Scholar
  21. 21.
    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993. Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Dip.to di Matematica “U. Dini”FirenzeItaly
  2. 2.Dip.to di Matematica “G. Vitali”ModenaItaly

Personalised recommendations