Dar’s conjecture and the log–Brunn–Minkowski inequality

Abstract

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 Boroczky, 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.