Danila Cherkashin,
IMI BAS

Abstract:

The most classical form of the Steiner tree problem is the following.

Problem 1. For a given finite set $P$ of points $x_1,\dots ,x_n\in {\mathbb{R}}^2$ to find a connected set $S$ with the minimal length (one-dimensional Hausdorff measure) containing $P$.

A solution to Problem 1 is called a Steiner tree; clearly it exists and it could be not unique. A remarkable and rarely known Oblakov theorem states that there is no pair of Steiner trees $S_1$ and $S_2$ which are codirected in terminals, i.e. $S_1\cap B_{\epsilon }(P)=S_2\cap B_{\epsilon }(P)$.

I present a new elementary and short proof and discuss corollaries of the Oblakov theorem and mathematics around this new proof which is of combinatorial topology flavor.