Firey P--and The Christoffel-the Minkowski Problem

In this paper we studied the existence and uniqueness of n-dimensional convex bodies in Rn+1 with prescribed k-th p--area measure ,where 1 k n.The p-area measures are defined corresponding to Firey's p-sum of convex bodies with p > I. The k = n case has been treated by some authors such as E.Lutwak, P.Guan and C.S.Lin, etc. Our problem is a generalization of the classical Christoffel-Minkow problem. We treated the this problem by seeking for the positive convex solutions of the associated fully nonlinear elliptic equations.First we proved the existence and uniqueness of the admissible solutions by the method of continuity and approximation; then by a deformation lemma we obtained the convex solutions under suitable assumption. Therefore, we solved the problem for p k + 1 under reasonable "convex assumption". Our results about the fully nonlinear elliptic equations may have its independent interest.