The Generalization And Application Of Several Famous Measure-inequalities Under The Brunn-minkowski Theory

This article belongs to the domian, is a high-speed developing geometry branch on the decade of late, of the Lp-Brumm-Minkowski theory. This M.D. disser-tation sketches firstly the growing history, researching status quo, main represent figures, and works of mathematicians of our country in the researching branch; The following, it studies the properties of the dual mixed p-affine surface area in Lp Space by applying the basic notions, basic theories and analysis inequalities of the Lp-Brunn-Minkowski theory. Then we find the more general forms of the famous Blaschke-Santalo inequality and Urysohn inequality and studies the mean width of Lp-radial linear combinations of convex bodies.The research works of this thesis consists of three parts.In the first part, we generalize the definition of mixed p-affine surface area, and introduce the concept of dual mixed p-affine surface area and definition of the dual mixed p-affine surface area ratio. Finally, we establish the affine isoperimetric inequality and dual Brumm-Minkowski inequality, and analyze their monotonicity.In the second part, after giving the definition of the dual mixed quermassinte-gral and Lp-mixed quermassintegral, we find the Blaschke-Santalo inequality and Urysohn inequality of them. Then using the general results of the two famous inequalities, we give some helpful results. In the third part, combining with the def-inition of the radial linear combinations, we introduce the concept of the Lp-radial linear combinations of convex bodies, and find the relations between the dual mixed volume of the convex bodies. Then we study the properties of the mean width of the Lp-radial linear combinations of convex bodies. At the same time, we find the infimum of the Lp-radial linear combinations of convex bodies and their polars.