John Ellipsoid Type Extremal Position Problems In Convex Geometric Analysis

The topic of this thesis belongs to convex geometric analysis.One of the im-portant questions in convex geometry is the affine extremal position problem,i.e.to find the extremal position of a convex geometry under linear transformations(usually SL(n)).Such positions contain the well-known John's ellipsoid,Lowner ellipsoid,Legendre ellipsoid,LYZ(Lutwak,Yang,Zhang)ellipsoid and their various generalizations.On the basis of the predecessors,we continue the investigation on affine extremal position problems.Precisely,we study the following two problems:(1)Minimal Orlicz mean width positions of convex bodies and related isoperi-metric inequalities.The main result is that denote by ? a convex,strictly increasing function and K C Rn be a convex body.then K be in the minimal Orlicz mean width position if and only if measure d??(K,·)= ?'(hK)hKda is isotropic.where hK(u)is the support function of K;d? is the rotationally invari-ant probability measure on Sn-1;(2)The extreme position of the relative Lp-eccentricity of a general spherical Borel measure.The Lp-eccentricity is a geometric quantity associated with the affine image of a general spherical Borel measure,it is usually used to indicate the magnitude of the difference in the linear transformation applied to the Borel measure of the sphere relative to the other transformation.It is of great significance to study its affine extremal position problems.In this thesis,we mainly study if p>0,what is the necessary and sufficient condition for the LP-eccentricity to obtain the minimum value under its affine transformation?...