No abstract uploaded!

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.

No abstract uploaded!

In this paper, a dual Orlicz–Brunn–Minkowski theory is presented. An Orlicz radial sum and dual Orlicz mixed volumes are introduced. The dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality are estab-lished. The variational formula for the volume with respect to the Orlicz radial sum is proved. The equivalence between the dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality is demonstrated. Orlicz intersection bodies are defined and the Orlicz–Busemann–Petty problem is posed.

The original goal of this paper is to extend the affine isoperimetric inequality and Steiner type inequality of Orlicz projection bodies (which originated to Lutwak, Yang, and Zhang [36]), from convex bodies to Lipschitz star bodies (whose radial functions are locally Lipschitz). In order to achieve it, we investigate the graph functions of the given Lipschitz star body K: Along almost all directions u, we can define the graph functions on an open dense subset of the orthogonal projection of K onto u⊥.