The space of parallel linear networks with a fixed boundary

In this paper we study the structure of the set $[G,\varphi]_\Gamma$ of immersed linear networks in $\mathbb R^N$ that are parallel to a given immersed linear network $\Gamma\colon G\to\mathbb R^N$ and whose boundary $\varphi$ coincides with the boundary of $\Gamma$. We prove that $[G,\varphi]_\Gamma$ is a convex polyhedral subset in the configuration space of moving vertices of the graph $G$. We also calculate the dimension of this convex subset and estimate the number of its faces of maximal dimension. The results obtained are used to describe the space of all locally minimal (weighted minimal) networks in $\mathbb R^N$ with a fixed topology and a fixed boundary. In the case of planar networks in which the degrees of vertices are at most three (Steiner networks), this dimension is calculated in topological terms.