The problem discussed in this paper was stated by Alexander O. Ivanov and
Alexey A. Tuzhilin in 2009. It stands at the intersection of the theories of
Gromov minimal fillings and Steiner minimal trees. Thus, it can be considered
as one-dimensional stratified version of the Gromov minimal fillings problem.
Here we state the problem; discuss various properties of one-dimensional
minimal fillings, including a formula calculating their weights in terms of
some special metrics characteristics of the metric spaces they join (it was
obtained by A.Yu. Eremin after many fruitful discussions with participants of
Ivanov-Tuzhilin seminar at Moscow State University); show various examples
illustrating how one can apply the developed theory to get nontrivial
results; discuss the connection with additive spaces appearing in
bioinformatics and classical Steiner minimal trees having many applications,
say, in transportation problem, chip design, evolution theory etc. In
particular, we generalize the concept of Steiner ratio and get a few of its
modifications defined by means of minimal fillings, which could give a new
approach to attack the long standing Gilbert-Pollack Conjecture on the
Steiner ratio of the Euclidean plane.