Integral Inequalities Of Curvature For Ovaloids In R~n And The Reverse Bonnesen-style Inequalities

Classical isoperimetric inequality describes the relation between two intrinsic invariants,which are the area A and the volume V of a convex body K.This paper researches the relation between the integration of curvature for the bound(?)K of the oval domain K and the surface area and the volume of K.By the Gauss curvature of the given ovaloid S in Rn,we construct a star body associated to the ovaloid S,called the star body by the Gauss curvature star body.We research the relation between the volume of the Gauss curvature star body and the area of the given ovaloid S and the volume enclosed by S,and we obtain some integral inequalities of curvature for ovaloid in the Euclidean space Rn.We also give the upper bound of the integral inequality about the mean curvature,and the lower bound of integral of ?h on the ovaloid,where ? is the Laplacian operator,his the support function of the ovaloid.At last,we give some reverse Bonnesen-type inequalities.