The Research On The Inequalities Of The Star Bodies And The Characteristics Of The Generalized Skew Hermite Matrix

This paper mainly studies two contents:the first is the inequality problem of the star inL_p-Brunn-Minkowski theory,the second is the generalized skew Hermite matrix problem.The inequality of the stars is a valuable problem in Brunn-Minkowski's theory,which lays a foundation for solving the extreme problems such as surface area,volume and width of convex bodies or stars.Star inequality has been widely used in random geometry,information theory,quantitative economics and stereology.In this paper,the Brunn-Minkowski inequality of dual quermassintegral is further studied by usingL_p-dual Brunn-Minkowski theory and method,Minkowski integral inequality,and the p-quermassintegral of Fiery linear combination andL_p-radial Minkowski linear combination of Brunn-Minkowski inequality in the Brunn-Minkowski theory is studied.This paper obtain the isolated form of the Brunn-Minkowski inequality for the p-quermassintegral of the Fiery linear combination and the dual quermassintegral of the L_p-radial Minkowski linear combination,and for the Minkowski inequality of the dual mixed quermassintegral in the Brunn-Minkowski theory.This paper also give the reinforcement form of Minkowski inequality for dual mixed p-quermassintegral and L_p-dual mixed quermassintegral.The generalized skew Hermite matrix is a matrix of special properties in matrix analysis,which is based on the study of Hermite matrix.The study of the skew Hermite matrix greatly enriches and expands the research results of the Hermitet matrix.The generalized skew Hermite matrix plays an important role in matrix theory,information theory,economic mathematics,cybernetics,linear system theory and many other fields.This paper study the characteristics of the generalized skew Hermite matrix,and give some equivalence conditions of the skew Hermiteian matrix.