# On the Configuration Space of Steiner Minimal Trees

## TL;DR

This paper proves that in Euclidean space, two finite point sets with unique, combinatorially equivalent Steiner minimal trees can be continuously deformed into each other while preserving their minimal tree structure.

## Contribution

It establishes a homotopy result for Steiner minimal trees, showing structural stability under continuous deformations in Euclidean space.

## Key findings

- Homotopy exists between point sets with equivalent Steiner minimal trees.
- Maintains the uniqueness and combinatorial structure during deformation.
- Provides insights into the stability of Steiner minimal trees.

## Abstract

Among other results, we prove the following theorem about Steiner minimal trees in $d$-dimensional Euclidean space: if two finite sets in $\mathbb{R}^d$ have unique and combinatorially equivalent Steiner minimal trees, then there is a homotopy between the two sets that maintains the uniqueness and the combinatorial structure of the Steiner minimal tree throughout the homotopy.