On Brunn-minkowski Inequality Of L_p -moment Body

Classical Brunn-Minkowski theory originated in Hermann Brunn's papers in 1887 and Hermann Minkowski's substantial part of the groundbreaking works.Its essence is the notes of mixed volume and basic Brunn-Minkowski inequalities.Lp-Brunn-Minkowski theory originated in Firey Lp-combinations(also known as Firey linear combination) of the convex bodies was defined by Firey in 1962,establishing this theory attributed to the famous mathematician E.Lutwak. In 1993,Lutwak introduced Firey Lp-combination of the convex bodies into the classical Brunn-Minkowski theory and proposed the concepts of Lp-mixed volume,Lp-mixed quermassintegrals,Lp-surface measure and Lp-mixed surface measure, and the corresponding integral expressions.Then the classical Brunn-Minkowski theory was extended into Lp-space research. In this paper,we combined the Blaschke Lp-harmonic combination and established the Brunn-Minkowski inequalities for the quermassintegrals and dual quermassintegrals of Lp-moment body. At the same time, we studied the monotonicity of Lp-moment body, and obtained monotone inequalities for quermassintegrals and dual quermassintegrals of Lp-moment body. In the mean time,we studied the monotonicity of the nonsymmetric Lp-moment bodies,and estabished some inequlities associated with their volume.The first chapter of this thesis, a brief history of disciplinary development and research status were given,and the main results were given.In the second chapter, we give the classical isoperimetric inequality and its basic theories, and then ob-tained the isoperimetric inequalities for the centroid and projection bodies.In the third chapter,we focus on the Brunn-Minkowski inequality of Lp-moments body. We combined the Blaschke Lp-harmonic combination and established the Brunn-Minkowski inequalities for the quermassintegrals and dual quermassintegrals of Lp-moment body. Then we defined the concepts of Lp-mixed moment body and Lp-mixed projection body, further research to establish some inequalities on the monotonicity of the quermassintegrals and dual quermassintegrals of Lp-(mixed) moment.In the fourth chapter,we analyzed nonsymmetric Lp-moment body.