Monotonicity Of Several Geometries In Complex Vector Spaces

The research content of this thesis belongs to L_p-Brunn-Minkowski theory and dual Brunn-Minkowski theory in convex geometric analysis.It focuses on several geometric bodies in the two theories,including general complex L_pmixed projection bodies,general complex L_pmixed centroid bodies,complex L_pmixed intersection bodies and related monotonicity inequalities.The specific research results are as follows:1.Based on the definition of general complex L_pprojection bodies in n-dimensional com-plex vector space given by Wang and Liu,the definition of general complex L_pmixed projection bodies is proposed by extending the L_pmixed volume of two convex bodies to the L_pmixed quermassintegral of two convex bodies.From the inequality of L_pmixed quermassintegrals of two convex bodies,the inequality of quermassintegrals of general complex L_pmixed projection bodies is derived,i.e.,the monotonicity of general complex L_pmixed projection bodies is proved.2.Since complex L_pcentroid bodies and complex L_pmoment bodies are dilates,using the definition of general complex L_pmoment bodies in n-dimensional complex vector space given by Wang and Liu,the definition of general complex L_pcentroid bodies is proposed,and then the definition of general complex L_pmixed centroid bodies is obtained.From the inequality of L_pmixed dual quermassintegrals of two convex bodies,the inequality between the quermassintegral of general complex L_pmixed centroid bodies is derived,i.e.,the monotonicity of general complex L_pmixed centroid bodies is proved.3.Based on the inequality of the L_pmixed(n-1)-dimensional dual quermassintegral for two intersection bodies in n-dimensional real vector space given by Yu,we extend this inequality to n-dimensional complex vector space,i.e.,the monotonicity of complex L_pmixed intersection bodies is proved.