Study On The Special Solution Of Christoffel-minkowski Problem

One of the core problems in the theory of Brunn-Minkowski is the characterization of convex bodies,which is an interdisciplinary subject in the fields of partial differential equations and convex geometric analysis.It has important applications in mathematics,physics,information theory and other disciplines.The Christoffel-Minkowski problem is that how to characterize the area measures of every order of convex bodies,which is a very important research problem in the theory of Brunn-Minkowski.In convex geometry analysis,the Christoffel-Minkowski problem is that,for 1?k?n-1,under what sufficient and necessary conditions,the Borel measure ? defined on the unit sphere S^n-1is the k order area measure of a convex body.Namely,whether there is a convex body K,s.t.?=S_k(K,·).In differential geometry,the Christoffel-Minkowski problem becomes that,given the funtion?(u)defined on S^n-1,whether there exists a convex body K,s.t.?(u)=s_k(u)=(?)R_i1(u)…R_ik(u),where{R₁(u),…,R_n-1(u)}are the principal curvature radius of the boundary point of the convex body K with u?S^n-1as the outer normal vector,and s_k(u)is the k-elementary symmetric function of principal curvature radius of K.The Christoffel-Minkowski problem has always been an open problem,which has been discussed by many famous mathematicians such as A.D.Aleksandrov,H.Buse-mann,Shiing-Shen Chern and so on.The necessary and sufficient conditions for the existence of convex solutions to the Christoffel-Minkowski problem of smooth convex revolution bodies are obtained by W.J.Firey.The sufficient conditions for the ex-istence of convex solutions are obtained by Professor P.F.Guan and X.N.Ma.In this paper,after we systematically understand the results by Firey,we obtain some necessary conditions for the Christoffel-Minkowski problem of convex revolution body in the sense of measure.