A convexification of the mailing version of the finite Gilbert problem for optimal networks is introduced. It is ia convex functional on the set of probability measures subject to the Wasserstein $p-$ metric.
The minimizer of this convex functional is a measure supported in a graph. If this graph is a tree (i.e contains no cycles) then this tree is also a minimum of the corresponding mailing Gilbert problem. The convexification of the Steiner problem is the limit of these convexified Gilbert's problems in the limit $p\rightarrow\infty$.