Lp Blaschke Addition And Related Extremal Problems

This dissertation deals with topics in convex geometry theory, among whichthe core of the theory is Brunn-Minkowski theory. This dissertation is devotedto study extreme problems in LpBrunn-Minkowski, including the LpBlaschkeaddition, Optimal afne Sobolev norm, complex slicing problem.The research work of this dissertation can be mainly divided into three as-pects.(1) We propose the notion of LpBlaschke for polytopes (1<p=n), whichgeneralize Lutwak’s definition for LpBlaschke of symmetric convex bodies. Sim-ilar to Lutwak’s proof, we establish the LpKneser-Su¨ss inequality for polytopes.Moreover, we show the coefcient of this inequality is best. Using this inequality,we can compare the volume of two polytopes by comparing their Lpsurface areas.(2) We study the optimal afne Sobolev norms. This research can be con-sidered as the extension work of Lutwak, Yang, Zhang on Lpafne Sobolev in-equality. Using the functional Minkowski problem for W1,p(Rn) due to Lutwak,Yang, Zhang, our problem is connected to isotropic p-surface area proposed byYu. In particular, we obtain several reverse afne Sobolev inequality by the studyof isotropic p-surface area.(3) We propose the concept of complex isotropic bodies. Furthermore, westudy some properties of complex isotropic body, and show the class of complexisotropic bodies is a larger than ones of real isotropic bodies. Similar to theresults in real vector space, we establish the relation between the complex slicingproblem and the complex isotropic constant of a body. From this relation, we obtain the extremal volume of complex central hyperplane section of the unit ballofnp(C)(p>1).