On the Regularity of the Solution of the n-Dimensional Minkowski Problem

where xi are the coordinate functions on S". Minkowski then asked the converse of the problem. Namely, given a positive function K defined on S" satisfying the above integral conditions, can we find a closed strictly convex hypersurface whose curvature function is given by K? Minkowski solved the problem in the category of polyhedrons. Then A. D. Alexandrov and others solved the problem in general. However, this last solution does not provide any information about the regularity of the (unique) convex hypersurface even if we assume K is smooth. In the two-dimensional case, H. Lewy was the first one who proved that if K is analytic, the solution to the Minkowski problem is also analytic. Around 1953, A. V. Pogorelov [9] and L. Nirenberg [6] solved the regularity problem in the smooth category independently. Their methods were quite different and restricted only to two dimensions. The method of Pogorelov was to show that the (unique) generalized solution of Alexandrov is smooth. This depends on the solvability and regularity of the Dirichlet problem of the two-dimensional Monge-Ampkre equation. The method of L. Nirenberg was to use the continuity method to produce a smooth solution directly. This depends on a priori estimates which are available only for elliptic equations of two variables.