We first characterize a polytope we prove some properties for the operator Г-2 whose new ellipsoid is a ball. Furthermore 2 and obtain some inequalities.
In this paper, by using the Lp-Brunn-Minkowski theory and its dual theory, L2-version on the conjectured projection inequality is investigated, the （reverse） inclusive relationship between L2-projection body and the classical projection body are established, and a constrained minimization problem is solved.
As a generalization of RpK, the mixed radial p-th mean bodies Rp（K,Lμ） have many properties similar to RpK. In this paper, some properties of Rp（K, L,μ） such as monotonicity, transform properties under volume-preserving linear transformations, etc. were studied. By using the polar coordinate formula for volume and Fubini＇s theorem, V（Rn（K, L,μ）） = V（L） was obtained.
