Steiner trees with infinitely many terminals on the sides of an angle

TL;DR

This work studies the Euclidean Steiner problem for an infinite, self-similar input placed on the two sides of a plane angle, defined by $A_{α,λ}$ with coordinates $(λ^{k-1}\cos α, ±λ^{k-1}\sin α)$. Under a small-angle and small-scale regime, the authors show every Steiner tree for the countable input $\mathcal{A}_1$ decomposes into a countable union of 5-terminal full* trees, and deduce an explicit length formula for these configurations; extending the input by adding a segment $[A_0B_0]$ yields exactly two indecomposable Steiner trees, each the mirror image of the other, with a second explicit length formula. The analysis employs Maxwell-type length formulas, Melzak reductions, and convexity arguments, and reveals a piecewise contraction dynamics governing the self-similar construction, linking geometric structure to a one-dimensional dynamical system. These results illuminate how infinite, boundary-attached terminal sets can induce decomposable versus indecomposable Steiner structures and open avenues for higher-dimensional analogues, density questions, and broader combinatorial classifications of optimal Steiner trees.

Abstract

The Euclidean Steiner problem is the problem of finding a set $St$, with the shortest length, such that $St \cup A$ is connected, where $A$ is a given set in a Euclidean space. The solutions $St$ to the Steiner problem will be called Steiner sets while the set $A$ will be called input. Since every Steiner set is acyclic we call it Steiner tree in the case when it is connected. We say that a Steiner tree is indecomposable if it does not contain any Steiner tree for a subset of the input. We are interested in finding the Steiner set when the input consists of infinitely many points distributed on two lines. In particular we would like to find a configuration which gives an indecomposable Steiner tree. We consider a self-similar input, namely the set $A_{α,λ}$ of points with coordinates $(λ^{k-1}\cos α,$ $\pm λ^{k-1}\sin α)$, where $λ>0$ and $α>0$ are small fixed values. These points are distributed on the two sides of an angle of size $2α$ in such a way that the distances from the points to the vertex of the angle are in a geometric progression. To our surprise, we show that in this case the solutions to the Steiner problem for $A_{α,λ}$, when $α$ and $λ$ are small enough, are always decomposable trees. More precisely, any Steiner tree for $A_{α,λ}$ is a countable union of Steiner trees, each one connecting 5 points from the input. By considering only a finite number of components we obtain many solutions to the Steiner problem for finite sets composed of $4k+1$ points distributed on the two lines ($2k+1$ on a line and $2k$ on the other line). These solutions are very similar to the ladders of Chung and Graham.

Figures (8)

• Figure 1: The left part contains two Steiner trees connecting the vertices of a square; the right part illustrates an example of a self-similar solution $\Sigma(\lambda)$.
• Figure 2: The input sets $\mathcal{A}_1=\{A_\infty, A_1,B_1,A_2,B_2,\dots\}$, $\mathcal{A}_0=\mathcal{A}_1\cup [A_0 B_0]$. Here $\alpha = \frac{\pi}{36}$ and $\lambda = \frac{1}{2}$.
• Figure 3: One of the solutions to the Steiner Problem for $\mathcal{A}_1$. See Theorem \ref{['theo:main2']}. Here $\alpha = \frac{\pi}{36}$ and $\lambda = \frac{1}{2}$.
• Figure 4: One of the two Steiner trees ${\mathcal{S}t}_0^1$, ${\mathcal{S}t}_0^2$ for the set $\mathcal{A}_0$, see Theorem \ref{['theo:main']}. The other solutions is the reflection with respect to the angle bisector. Here $\alpha = \frac{\pi}{36}$ and $\lambda = \frac{1}{2}$.
• Figure 5: Definition of the point $W_1$ in Lemma \ref{['lm:321066']}

Theorems & Definitions (43)

• Theorem 1
• Corollary 1
• Theorem 2
• Remark 1
• Theorem 3
• Lemma 1
• Corollary 2
• Remark 2
• Lemma 2: Maxwell-type formula
• Remark 3