Research Of The Extremal Theory In Convex Geometrical Analysis

The researches of this thesis belong to Brunn-Minkowski theory,which is a high-speed developing geometry branch and is an important brand of geometric.This thesis devotes to the study of functional inequalities and extremal problems by using theory of geometry analysis.In chapter 3,we characterize the minimal??and^°??for any?SL?9?)using isotropic measures on the Euclidean unit sphere.The condition for the minimum of??^°??is also obtained.In dual Brunn-Minkowski theory,we introduce the notion of dual-difference and study its several properties.Next we get volume inequality of dual-difference using dual Brunn-Minkowski inequality.Moreover,we also give notion of dual-parallel bodies.Convexity and continuity of this new family are proved.In order to study convexity of dual-parallel bodies,we also define harmonic-mean of two real numbers.In chapter 5,we introduce the notion of asymmetric Orlicz difference body and study its properties systematically.Next we give the asymmetric Orlicz difference body and the Minkowski addition of convex bodies are closely related.Finally,we study the extremal values of the volumes.In the course of studying extreme values,we useBrunn-Minkowski inequality and the properties in Orlicz setting.In chapter 6,we major study Poincar?e-type inequality on the Euclidean unit sphere⁹-1).Firstly,we give a selfadjoint operator.Secondly,we get the first variational formula and the second variational formula using this operator.Finally,we get main research results by using two variational formulas.