Bonnesen Symmetric Mixed Equivalent Inequalities And L_p Mixed Masses

The isoperimetric problem is the most classical and the most important problem in geometry and convex geometric analysis. The isoperimetric inequality is one of the most important inequalities in geometry and analysis, which is equivalent to Sobolev inequality in analysis. The Bonnesen-style inequalities is to promote and strengthen isoperimetric inequality. In recently, the Bonnesenstyle inequalities has been generalized to the plane of constant curvature. There is the difficult problem in integral geometry and geometric inequality for studying the Bonnesen-style inequalities in the higher dimension Euclidean space Rn. Even so, some progress has been made about the issue recently.In this dissertation, we study the another extension of the isoperimetric inequality and the Bonnesen-style inequality in the Euclidean plane R~2. That is,we study the Minkowski inequality for two convex domains and the Bonnesenstyle(Minkowski) symmetric mixed isohomothetic inequality in R~2. We estimate the translative containment measure for a convex domain to contain,or to be contained in, the homothetic copy of another convex domain in the Euclidean plane R~2. We estimate the symmetric mixed isohomothetic deficit?2(K0, K1) ≡ A201- A0A1(where A0, A1 be areas of two convex domains K0, K1,respectively, in R~2, A01 be the mixed area of K0 and K1). We obtain a sufficient condition for a convex domain to contain, or to be contained in, the homothetic copy of another convex domain in R~2. We obtain some the Bonnesen-style symmetric mixed isohomothetic inequalities and reverse Bonnesen-style symmetric mixed isohomothetic inequalities. We obtain the homothetic Bol-Fujiwara theorem. We study the L_pmixed centroid body about convex bodies K1,..., Knin the n-dimension Euclidean space Rn. We obtain some geometric inequalities of the L_pmixed centroid body. We obtain some the latest results in the dissertation.In Chapter 3, we study the translative containment measure. Via the translative kinematic formulas of Poincar′e and Blaschke in integral geometry, we investigate the translative containment measure for a convex domian to contain, orto be contained in, the homothetic copy of another convex domian in the Euclidean plane R~2. We obtain the homothetic containment measure theorem and translative containment measure theorem.In Chapter 4, we estimate the upper bounds and lower bounds of the symmetric mixed isohomothetic deficit about two plane convex domians. At first,we define the inradius and outradius of a convex domain with respect to another convex domain. We obtain some Bonnesen-style symmetric mixed isohomothetic inequalities by translative containment measure theorem. Especially, these inequalities obtained are known Bonnesen-style isopermetric inequalities if one of domains is a disc. Secondly, we define the curvature inradius and curvature outradius of a oval domain with respect to another oval domain. We obtain some reverse Bonnesen-style symmetric mixed isohomothetic inequalities by translative containment measure theorem. These inequalities obtained are known the reverse Bonnesen-style isopermetric inequalities if one of domains is a disc. As direct consequences, we obtain these symmetric mixed isohomothetic inequalities in the dissertation that strengthen the known Minkowski inequality for mixed areas in the Euclidean plane R~2. We obtain the homothetic Bol-Fujiwara theorem.In Chapter 5, we study L_pmixed centroid body. We define the L_pmixed centroid body Γp(K1,..., Kn) for convex bodies K1,..., Knwith containing the origin in their interiors in the n-dimension Euclidean space Rn. We obtain some important inequalities of the L_pmixed centroid body Γp(K1,..., Kn).