The Inequality Of Brunn-Minkowski Of John Ellipsoid Study

Integral geometry is the origin of the development of convex geometries.The Brunn-Minkowski mixed volume theory is the central theory of convex geometries.The core of the theory is the classical Brunn-Minkowski inequality.The John ellipsoid is a major approximation of the convex body,and the properties of many convex bodies can be accomplished by the John ellipsoid.Therefore,the John ellipsoid is a basic tool in the convex geometries.and the John ellipsoid is one class of very important special convex body of the convex geometries and functional analysis classic and.In the optimization of theory,statistics and related computer technology and other fields are used widely.Based on the John ellipsoid theorem and the classical Brunn-Minkowski theory,this paper focuses on the transformation of the convex body in the n-dimension Euclidean space R~n:the convex body K transformed to the John ellipsoid JK of convex body K.On the basis of this transformation,it is pointed out that the Brunn-Minkowski type inequality of the John ellipsoid of the convex body is established.Finally,according to the Brunn-Minkowski theory,the Brunn-Minkowski type inequality of the John ellipsoid of the convex body is extended to the meaning of L_p sum.