Convex Bodies: The Brunn-Minkowski Theory

This article belong to the Brunn-Minkowski theory, which is a high-speed developing geometry branch during the past several decades. This thesis works for theoretical study on isoperimetric problem and related inequalities by using theory of geometry analysis, way of integral transformations and analysis inequalities.The research works of this thesis consists of three parts.In classical Brunn-Minkowski theory, we establish an extension of the matrix form of the Brunn-Minkowski inequality. As applications, we give generalizations on metric addition inequality of Alexander.In dual Brunn-Minkowski theory, we study the properties of the dual harmonic quer-massintegrals systematically and establish some inequalities for the dual harmonic quer-massintegrals, such as the Minkowski inequality, the Brunn-Minkowski inequality, the Blaschke-Santalo inequality and the Bieberbach inequality. We establish the dual Brunn-Minkowski inequality for dual affine quermassintegrals. Recently we learned that Gardner have independently proved it by a different method. The polar body of a convex body is an important object in the context of convex geometry. Hence, after we studied the intersection bodies, it is natural to consider the inequalities for their polar bodies. But the intersection body of even a convex body generally is not convex. Thus the inequalities for the polar body of the intersection body can not be given in the general cases. Applying the concept of star dual, which is introduced by Moszynska, we establish some inequalities for star duals of intersection bodies. Hence, we present a new kind of duality between intersection bodies and projection bodies. At last, we introduce the dual mixed body and establish some properties and inequalities of it. As applications, we strength some inequalities in the the dual Brunn-Minkowski theory.In Brunn-Minkowski-Firey theory, we establish two extremum properties of the new ellipsoid; Then we generalize Petty's affine projection inequality and monotonicity results related to affine surface area to L_p—affine surface area; We establish the Brunn-Minkowski type inequalities for the volume of the L_p centroid body and its polar body with respect to the normalized L_p radial addition. At last, we introduce the minimal L_p—mean width of a convex body and generalize the minimal mean width to the Brunn-Minkowski-Firey theory. Furthermore, we get the stability version of the L_p mean width position for L_p projection body.