Translator Disclaimer
May 2016 Dar’s conjecture and the log–Brunn–Minkowski inequality
Dongmeng Xi, Gangsong Leng
J. Differential Geom. 103(1): 145-189 (May 2016). DOI: 10.4310/jdg/1460463565


In 1999, Dar conjectured that there is a stronger version of the celebrated Brunn-Minkowski inequality. However, as pointed out by Campi, Gardner, and Gronchi in 2011, this problem seems to be open even for planar $o$-symmetric convex bodies. In this paper, we give a positive answer to Dar’s conjecture for all planar convex bodies. We also give the equality condition of this stronger inequality.

For planar $o$-symmetric convex bodies, the log–Brunn–Minkowski inequality was established by Böröczky, Lutwak, Yang, and Zhang in 2012. It is stronger than the classical Brunn–Minkowski inequality, for planar $o$-symmetric convex bodies. Gaoyong Zhang asked if there is a general version of this inequality. Fortunately, the solution of Dar’s conjecture, especially, the definition of “dilation position”, inspires us to obtain a general version of the log–Brunn–Minkowski inequality. As expected, this inequality implies the classical Brunn–Minkowski inequality for all planar convex bodies.


Download Citation

Dongmeng Xi. Gangsong Leng. "Dar’s conjecture and the log–Brunn–Minkowski inequality." J. Differential Geom. 103 (1) 145 - 189, May 2016.


Published: May 2016
First available in Project Euclid: 12 April 2016

zbMATH: 1348.52006
MathSciNet: MR3488132
Digital Object Identifier: 10.4310/jdg/1460463565

Rights: Copyright © 2016 Lehigh University


Vol.103 • No. 1 • May 2016
Back to Top