A Problem of Alexandrov

0 Introduction For n 2, Let M n be a nite convex, not necessarily smooth, hypersur-face in Euclidean space R n+1 containing the origin. More precisely, M n is the boundary of some convex domain in R n+1 containing a neighborhood of the origin. We write M n = fR(x) = (x)x j x 2 S n g, where is a function from S n to R +. Let : M n ! S n denote the generalized Gauss map, namely, (Y) is the set of outward unit normals to supporting hy-perplanes of M n at Y. The integral Gaussian curvature of M n is deened by (F) = j(R(F))j; for all Borel set F S n : It is clear that is a nonnegative, completely additive function on the Borel sets of S n. For any set F S n , let Cone(F) = ftX j X 2 F; t 0g be the cone generated by F. For any cone C R n+1 , let C = fX 2 R n+1 j X Y 0; 8 Y 2 Cg be the dual cone. F = (Cone(F)) \ S n is the dual angle. The following theorem was obtained by A.D. Alexandrov in 1]. See also 2] and 24] for exposition. Theorem A A necessary and suucient condition for a nonnegative, completely additive function on the Borel sets of S n to be the integral Gaussian curvature of some nite convex hypersurface in Euclidean space R n+1 containing the origin is: (A1) (S n) = jS n j, (A2) For every convex subset F of S n , (F) < jS n j ? jF j. Such hypersurface is unique up to a homothetic transformation.