THE CHRISTOFEL-MINKOWSKI PROBLEM III: EXISTENCE AND CONVEXITY OF ADMISSIBLE SOLUTIONS

1. introduction This paper is a sequel to [15] on geometric fully nonlinear partial differential equations associated to the Christoffel-Minkowski problem. In [15], we considered the existence of convex solutions of the following equation: Sk(uij + uδij) = ϕ on Sn,(1.1) where Sk is the k-th elementary symmetric function and uij the second order covariant derivatives of u with respect to orthonormal frames on Sn, and where a function u ∈ C2(Sn) is called convex if (1.2) (uij + uδij)> 0, on Sn. It is known that (e.g., see [24, 11]) ∀v ∈ C2(Sn),∫ Sn xmSk(vij(x) + v(x)δij)dx = 0, ∀m = 1, 2,..., n + 1. A necessary condition for equation (1.1) to have a solution is∫ Sn xiϕ(x)dx = 0, ∀i = 1, 2,..., n + 1.(1.3) Condition (1.3) is also sufficient for the Minkowski problem, which corresponding to k = n in equation (1.1). In this case, equation (1.1) is the Monge-Ampère equation corresponding to the Minkowski problem: (1.4) det(uij + uδij) = ϕ on Sn. The Minkowski problem has been settled completely by Nirenberg [21] and Pogorelov [22] for in dimension 2 and by Cheng-Yau [5] and Pogorelov [24] for general dimensions. From their work, for any positive function ϕ ∈ C2(Sn) satisfying the necessary condition (1.3), Monge-Ampère equation (1.4) always has a convex solution