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A convexification of the mailing version of the finite Gilbert problem foroptimal networks is introduced. It is ia convex functional on the set ofprobability measures subject to the Wasserstein $p-$ metric. The minimizer ofthis convex functional is a measure supported in a graph. If this graph is atree (i.e contains no cycles) then this tree is also a minimum of thecorresponding mailing Gilbert problem. A numerical algorithm for theimplementation of the convexified Gilbert-mailing problem is also suggested,based on entropic regularization.

Is the mailing Gilbert-Steiner problem convex?