Gromov minimal fillings for finite metric spaces

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 mini- mal 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 partic- ipants of Ivanov-Tuzhilin seminar at Moscow State University); show various examples illustrating how one can apply the developed theory to get non- trivial results; discuss the connection with additive spaces appearing in bioin- formatics 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 at- tack the long standing Gilbert-Pollack Conjecture on the Steiner ratio of the Euclidean plane.