On Researches Of Geometric Inequalities In Convex Geometry Analysis

This paper belongs to the convex geometric analysis theory, which devoted to studying general Lp-brightness integral, Lp-dual mixed geominimal surface area, Lp-cross-section, complex intersection. The main technologies used in this paper are basic concepts, basic methods and integral transforms in the Brunn-Minkowski theory and its dual theory.The main results can be stated as follows:1. Combining with general Lp-projection bodies and brightness integral, we define general Lp-mixed brightness integrals, and research their properties and extremal-values problem and some geominimal inequalities.2. Combining with the definition of Lp-mixed geominimal surface area and integral formula of Lp-mixed geominimal surface area, we give the integral formula of Lp-dual mixed geominimal surface area and establish some inequalities; Based on the Lp-mixed geominimal surface area for multiple convex bodies, we define the concept of Lp-dual mixed geominimal surface area for multiple star bodies( p 1-n) and establish some inequalities related to this concept.3. Based on the concept of Lp-cross-section body, we further research the Lp-cross-section and establish some inequalities for dual quermassintegrals of this concept.4. Combining with the notions of complex intersection bodies and two combinations of complex star bodies, we further research complex intersection bodies and establish some Brunn-Minkowski type inequalities of complex intersection bodies for dual quermassintegrals.