Embedding Sn into Rn+1 with given integral Gauss curvature and optimal mass transport on Sn

In [A.D. Aleksandrov, Convex Polyhedra, GITTL, Moscow, USSR, 1950 (in Russian); English translation: A.D. Aleksandrov, Convex Polyhedra, Springer, Berlin–New York, 2005] A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gauss curvature. Our solution includes the case of a convex polytope. This problem was also first considered by Aleksandrov and below it is referred to as Aleksandrov's problem. The second goal of this paper is to show that in variational form the Aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations.