Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Böröczky, K.J., Lutwak, E., Yang, D., Zhang, G.: The log-Brunn–Minkowski inequality. Adv. Math. 231, 1974–1997 (2012)
Chavel, I.: Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives. Cambridge University Press, Cambridge (2001)
Dergiades, N.: An elementary proof of the isoperimetric inequality. Forum Geom. 2, 129–130 (2002)
Gage, M.E.: An isoperimetric inequality with applications to curve shortening. Duke Math. J. 50, 1225–1229 (1983)
Gage, M.E.: Curve shortening makes convex curves circular. Invent. Math. 76, 357–364 (1984)
Gage, M.E.: Evolving plane curves by curvature in relative geometries. Duke Math. J. 72, 441–466 (1993)
Gage, M.E., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986)
Green, M., Osher, S.: Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves. Asian J. Math. 3, 659–676 (1999)
Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238 (1978)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, Second expanded edn. Cambridge University Press, Cambridge (2014)
Xi, D.M., Leng, G.S.: Dar’s conjecture and the log-Brunn–Minkowski inequality. J. Differ. Geom. 103, 145–189 (2016)
Yang, Y.L., Zhang, D.Y.: The Green–Osher inequality in relative geometry. Bull. Aust. Math. Soc. 94, 155–164 (2016)
Yang, Y.L., Zhang, D.Y.: The log-Brunn–Minkowski inequality in \(\mathbb{R}^3\). In: To appear in Proceedings of the AMS. (2018). https://doi.org/10.1090/proc/14366
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author is supported by the Doctoral Scientific Research Foundation of Liaoning Province (No. 20170520382) and the Fundamental Research Funds for the Central Universities (No. 3132017046).
Rights and permissions
About this article
Cite this article
Yang, Y. Nonsymmetric extension of the Green–Osher inequality. Geom Dedicata 203, 155–161 (2019). https://doi.org/10.1007/s10711-019-00430-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-019-00430-8