Nonsymmetric extension of the Green–Osher inequality

  • Yunlong YangEmail author
Original Paper


In this paper we obtain the extended Green–Osher inequality when two smooth, planar strictly convex bodies are at a dilation position and show the necessary and sufficient condition for the case of equality.


Dilation position Green–Osher’s inequality Nonsymmetric Relative Steiner polynomial 

Mathematics Subject Classification 2010

52A40 52A10 



I am grateful to the anonymous referee for his or her careful reading of the original manuscript of this paper and giving us many invaluable comments. I would also like to thank Professor Shengliang Pan for posing this problem to me.


  1. 1.
    Böröczky, K.J., Lutwak, E., Yang, D., Zhang, G.: The log-Brunn–Minkowski inequality. Adv. Math. 231, 1974–1997 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chavel, I.: Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  3. 3.
    Dergiades, N.: An elementary proof of the isoperimetric inequality. Forum Geom. 2, 129–130 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Gage, M.E.: An isoperimetric inequality with applications to curve shortening. Duke Math. J. 50, 1225–1229 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gage, M.E.: Curve shortening makes convex curves circular. Invent. Math. 76, 357–364 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gage, M.E.: Evolving plane curves by curvature in relative geometries. Duke Math. J. 72, 441–466 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gage, M.E., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Green, M., Osher, S.: Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves. Asian J. Math. 3, 659–676 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, Second expanded edn. Cambridge University Press, Cambridge (2014)zbMATHGoogle Scholar
  11. 11.
    Xi, D.M., Leng, G.S.: Dar’s conjecture and the log-Brunn–Minkowski inequality. J. Differ. Geom. 103, 145–189 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Yang, Y.L., Zhang, D.Y.: The Green–Osher inequality in relative geometry. Bull. Aust. Math. Soc. 94, 155–164 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yang, Y.L., Zhang, D.Y.: The log-Brunn–Minkowski inequality in \(\mathbb{R}^3\). In: To appear in Proceedings of the AMS. (2018).

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of ScienceDalian Maritime UniversityDalianPeople’s Republic of China

Personalised recommendations