  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. A numerical algorithm for the
implementation of the convexified Gilbert-mailing problem is also suggested,
based on entropic regularization.
