Extremal Problems For P-Mean Width Of Convex Bodies

This thesis is devoted to the mean width of convex bodies.As an extremely important metric of convex bodies,the mean width has attracted an increasing interest.Among them the most celebrated inequality is the classical Urysohn inequality (?)with equality if and only if K is a ball in Rn.Here we denote by w(K)the mean width of K,and wn the volume of the unit ball in Rn.The topics on Urysohn type inequalities,belong to the category of isoperimetric problem,whose equality conditions often are characterized by balls in Rn.As an aside,such iequalities usually can be proved by using Steiner's symmetrization.A harder aspect of research on the mean width of convex bodies lies in that if there exist a class of inequalities on mean width with non-balls os the characterizing convex body of the equalities.A trivial fact is that,such inequalities can t be established by employing Steiner s symmetrization.In this thesis,we carry on study on the p-mean width of convex bodies.In brief,we establish some inequalities with cube,octahedron and their variants as the equality characterizing bodies.To achieve this,we establish some weighted reverse isoperimet-ric inequalities of p-mean width of convex bodies.The main results in the thesis are theorem 1.2.1 and theorem 1.2.2.More precisely,the first class of such inequalities is a new connection between isotropic measure on unit sphere,p-mean width,and a notion of Lp-pseudo-moment body of the referred body.The equality is characterized by octa-hedron and its variants.Secondly,we provide the relationship between the integral of Gaussian measure on cubes and p-mean width of convex bodies.The equality condition is determined by cube and its variants.These results contribute the research on extremal problems of p-mean width of convex bodies.