The Existence of Convex Body with Prescribed Curvature Measures

Abstract

Curvature measure and surface area measure are the basic notions in the classical differential geometry. They play fundamental roles in the theory of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. The Minkowski problem is the problem of prescribing n-th surface area measure on Sn. The Christoffel problem concerns the prescribing the 1-st surface area measure (e.g., see [1, 14, 17, 6, 19, 7, 3]). The general problem of prescribing surface area measures is called the Christoffel-Minkowski problem, we refer [12] for an updated account. The problem of prescribing 0-th curvature measure is called the Alexandrov problem, which is a counterpart to Minkowski problem. The problem is equivalent to solve a Monge-Ampere type equation on Sn. The existence and uniqueness were obtained by Alexandrov [2]. The regularity of the Alexandrov problem in elliptic case was proved by Pogorelov [18] for n = 2 and by Oliker [16] for higher dimension case. The general regularity results (degenerate case) of the problem were obtained in [9]. The general problem of prescribing (n− k)-th curvature measure for case k ≤ n is an interesting counterpart of the Christoffel-Minkowski problem. It has been discussed in literature (e.g., [20]). Nevertheless, very little is known except for the Alexandrov problem. In this paper, we are concerned with the existence of convex bodies with the prescribed (n− k)-th curvature measure for 1 ≤ k < n. We start with the definitions of curvature measures and surface area measures for convex bodies with smooth boundary. Let Ω be a bounded convex body in Rn+1 with C2 boundary M , the corresponding curvature measures and surface area measures of Ω can be defined according to some geometric quantities of M . Let κ = (κ1, · · · , κn) be the principal curvatures of M at point x, let Wk(x) = Sk(κ(x)) be the k-th Weingarten curvature of M at x (where Sk is the k-th elementary symmetric function). In particular, W1,W2