Dar's conjecture and the log-Brunn-Minkowski inequality

Dongmeng Xi Shanghai University Gangsong Leng Shanghai University

Convex and Discrete Geometry mathscidoc:1703.40037

Distinguished Paper Award in 2017

Journal of Differential Geometry, 103, 145-189, 2016
In 1999, Dar conjectured if 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 B¨or¨oczky, 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.
Convex body, Brunn-Minkowski inequality, Dar’s conjecture, log-Brunn- Minkowski inequality, dilation position.
[ Download ] [ 2017-03-10 21:14:18 uploaded by xidongmeng ] [ 1363 downloads ] [ 0 comments ]
  title={Dar's conjecture and the log-Brunn-Minkowski inequality},
  author={Dongmeng Xi, and Gangsong Leng},
  booktitle={Journal of Differential Geometry},
Dongmeng Xi, and Gangsong Leng. Dar's conjecture and the log-Brunn-Minkowski inequality. 2016. Vol. 103. In Journal of Differential Geometry. pp.145-189. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170310211418681589653.
Please log in for comment!
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved