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Abstract

In his book on convex polytopes \cite{Al-pol} A. D. Aleksandrov raised a general question of finding variational formulations and solutions to geometric problems of existence of convex polytopes in ${\mathbb R}^{n+1}$, $n \geq 2,$ with prescribed geometric data. Examples of such problems for closed convex polytopes for which variational solutions are known are the celebrated Minkowski problem \cite{Al-pol} and the Gauss curvature problem \cite{Oliker:TAMS}. In this paper we give a simple variational proof of existence for the A. D. Aleksandrov problem \cite{Al:42, Al-pol} in which the hypersurface in question is a polyhedral convex graph over the entire ${\mathbb R}^n$, has a prescribed asymptotic cone at infinity, and whose integral Gauss-Kronecker 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 \cite{Gangbo:94} and L. Caffarelli \cite{Caf3}. The presented treatment of the Aleksandrov problem is self-contained and independent of the Monge-Kantorovich theory.