Extremum Problems For Convex Bodies

This article belong to the domain, which is a high-speed developing geometry branch on the several decades of late, of the Brunn-Minkowski theory. The development survey and main research directions of convex geometry are presented in the preface. This thesis deals with some inequalities and extreme problems by applying the basic notions, basic theories and integral transforms of the Brunn-Minkowski theory and the L_n-Brunn-Minkowski theory.We study some extreme problems in the Brunn-Minkowski theory. First, We discuss the relationship between the width integral of convex body and dual quermassintegrals when i∈R, and we obtain Some inequalities and extremum properties.Second, some inequalities are shown, among mixed affine surface areas, mixed projections and mixed volumes of centroid body. Moreover, the inequality analogous to the Blaschke-Santald inequality is established for the mixed affine surface areas. The general forms of the dual Urysohn inequalities are given.Third, we establish the Petty-Schneider Theorem for mixed body.In the aspects of the basic theory for Brunn-Minkowski theory, we first show the dual of the mixed width-integral of convex bodies -the notion of mixed chord-integrals of star bodies. We establish the Fenchel-Aleksandrov inequality and a general isoperimetric inequality for the mixed chord-integrals. Further, the dual general Bieberbach inequality is presented. As an application of the dual form, a Brunn-Minkowski type inequality for mixed intersection bodies is given.In the aspects of the inequalities and extremum properties of geometry bodies in the L_p-Brunn-Minkowski theory. For the L_p-projection body and L_p-centroid body, we mainly research the monotonicity inequalities of the L_p-projection body and L_p-centroid body and their polar.