Some new Brunn-Minkowski-type inequalities in convex bodies

The Brunn-Minkowski inequality theory plays an important role in a number of mathematical disciplines such as measure theory, crystallography, optimal control theory, functional analysis, and geometric convexity. It has many useful applications in combinatorics, stochastic geometry, and mathematical economics. In recent years, several authors including Ball [1, 2, 3], Bourgain and Lindenstrauss [7], Gardner [8, 9, 10], Schneider [25], Lutwak [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], have given considerable attention to the Brunn-Minkowski inequality theory and its various generalizations. The purpose of this paper is to establish some new-type BrunnMinkowski inequalities and Minkowski inequalities and their inverse versions. As an application, we generalize and improve some interrelated results. Notation and preliminary works are given in Section 2. In Section 3.1, some new inequalities of width-integrals, with equality conditions for convex bodies and their inverse versions are established. And we also establish some similar inequalities for projection bodies. This work generalizes a result which was given in [13]. Most importantly, we find the Brunn-Minkowski inequality for polars of mixed projection bodies in the section. Generalizations and inverse versions of the dual Minkowski inequality and BrunnMinkowski inequalities, with inequality conditions, for the radial Minkowski linear combination are presented in Section 3.2. These results generalize some results which were given by Lutwak [12, 17]. A general dual Brunn-Minkowski inequality, with equality conditions for radial Blaschke linear combination and its inverse version were given in Section 3.3. A special case of the result is just a new inequality, which was given by Lutwak [17]. A dual Brunn-Minkowski inequality for the harmonic Blaschke linear combination and its inverse version, with equality conditions, were established in Section 3.4.