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 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.

Quantitative anisotropic isoperimetric and Brunn−Minkowski inequalities for convex sets with improved defect estimates

“…Recall, that Dar's conjecture [12] asserts the following: For any bounded convex bodies, the inequality holds |K + L| 1/n ≥ M(K, L) 1/n + |K| 1/n |L| 1/n M(K, L) 1/n , where M(K, L) = max x∈R n |K ∩ (L + x)|. Note, that Dar's conjecture has been proven recently by Xi and Leng [38] in the planar case n = 2. Remark 3.4.…”

Nonsymmetric extension of the Green–Osher inequality

“…Noticing that there is a common center when K and L are at a dilation position, then the ratio of the support functions of K and L belongs to the range from r(K, L) to R(K, L), which leads to the Green-Osher inequality holds without symmetric condition. Properties of convex bodies are at a dilation position can be found in Lemma 3.1 (see also Xi-Leng [11]). In this paper, inspired by the impressive work in [11], we obtain the main result.…”

“…Properties of convex bodies are at a dilation position can be found in Lemma 3.1 (see also Xi-Leng [11]). In this paper, inspired by the impressive work in [11], we obtain the main result.…”

Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

“…In these papers the notions of -addition, -mixed volume, -affine surface area, -centroid body, andprojection body and inequalities were extended to an Orlicz setting. Advances in the theory and its dual can be found in [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48].…”