摘要
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.
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.
基金
supported by National Natural Science Foundation of China(Grant Nos.11271302
11161007 and 11401486)
the Ph.D.Program of Higher Education Research Fund(Grant No.2012182110020)
Guizhou Foundation for Science and Technology(Grant No.LKS[2011]6)
