601 research outputs found
Embedding into with given integral Gauss curvature and optimal mass transport on
In his book on Convex Polyhedra (section 7.2), 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
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