In this paper we exploit a slight variant of a result previously proved in [11] to define a procedure which delivers the convex envelope of some bivariate functions over polytopes. The procedure is based on the solution of a KKT system and simplifies the derivation of the convex envelope with respect to previously proposed techniques. The procedure is applied to derive the convex envelope of the bilinear function xy over any polytope, and the convex envelope of functions x^ny^m over boxes.