The Extremal Problems For The General Measure ? Of Convex Body

In this paper,we study the extreme problems of the general measure in Euclidean space R~n and some related properties of the inclusion measure of convex body.The existence theorem of the solution of the Orlicz-Minkowski problem for the general measure and the stronger boundedness of the inclusion measure of the convex body are obtained.The Minkowski problem is one of the hot topics in the field of convex geometry.The third chapter of this paper studies the Orlicz-Minkowski problem associated to the general measuremwith the homogeneous density,and proves the existence of the solution to the Minkowski problem whenmis the even measure.At the same time,we also study the extreme problems of the mixed measure of the convex body,and define the Orlicz geominimal measure associated with the general measure.Furthermore,we will use the Orlicz Petty body of the general measure to prove the continuity of the Orlicz geominimal measure.In the fourth chapter,we study some important properties of the boundedness of the inclusion measure of the convex bodies.By the technique such as the inner parallel body,we have established some new isoperimetric type inequalities,which give the upper and lower bounds of the measure.We also found the form body of convex body plays an important role in the boundedness of the inclusion measure.