  In this paper we give a new proof of a theorem by Alexandrov on the Gauss
curvature prescription of Euclidean convex sets. This proof is based on the
duality theory of convex sets and on optimal mass transport. A noteworthy
property of this proof is that it does not rely neither on the theory of convex
polyhedra nor on P.D.E. methods (which appeared in all the previous proofs of
this result).
