Translative containment measure and symmetric mixed isohomothetic inequalities

We first investigate the translative containment measure for convex domain K0 to contain, or to be contained in, the homothetic copy of another convex domain K1, i.e., given two convex domains K0, K1 of areas A0, A1, respectively, in the Euclidean plane R2, is there a translation T so that t（T K1）  K0 or t（T K1） ？ K0 for t 〉 0？ Via the translative kinematic formulas of Poincar′e and Blaschke in integral geometry,we estimate the symmetric mixed isohomothetic deficit σ2（K0, K1） ≡ A201- A0A1, where A01 is the mixed area of K0 and K1. We obtain a sufficient condition for K0 to contain, or to be contained in, t（T K1）. We obtain some Bonnesen-style symmetric mixed isohomothetic inequalities and reverse Bonnesen-style symmetric mixed isohomothetic inequalities. These symmetric mixed isohomothetic inequalities obtained are known as Bonnesen-style isopermetric inequalities and reverse Bonnesen-style isopermetric inequalities if one of domains is a disc. As direct consequences, we obtain some inequalities that strengthen the known Minkowski inequality for mixed areas and the Bonnesen-Blaschke-Flanders inequality.