A variational solution of the A.D. Aleksandrov problem of existence of a noncompact complete convex polytope with prescribed integral Gauss curvature

In his book on convex polytopes [2] A.D. Aleksandrov raised a general question of finding variational formulations and solutions to geometric problems of existence of convex polytopes in Rn+1, n ≥ 2, with prescribed geometric data. Examples of such problems for closed convex polytopes for which variational solutions are known are the celebrated Minkowski problem [2] and the Gauss curvature [18] problem. In this paper we give a simple variational proof of existence for the A.D. Aleksandrov problem [1, 2] in which the hypersurface in question is a polyhedral convex graph over the entire Rn, has a prescribed asymptotic cone at infinity, and whose integral GaussKronecker curvature has prescribed values at the vertices. The functional that we use is motivated by the functional arising in the dual problem in the Monge-Kantorovich optimal mass transfer theory considered by W. Gangbo [13] and L. Caffarelli [11]. The presented treatment of the Aleksandrov problem is self-contained and independent of the Monge-Kantorovich theory1.