  In the Euclidean Steiner Tree problem, we are given as input a set of points
(called terminals) in the $\ell_2$-metric space and the goal is to find the
minimum-cost tree connecting them. Additional points (called Steiner points)
from the space can be introduced as nodes in the solution.
  The seminal works of Arora [JACM'98] and Mitchell [SICOMP'99] provide a
Polynomial Time Approximation Scheme (PTAS) for solving the Euclidean Steiner
Tree problem in fixed dimensions. However, the problem remains poorly
understood in higher dimensions (such as when the dimension is logarithmic in
the number of terminals) and ruling out a PTAS for the problem in high
dimensions is a notoriously long standing open problem (for example, see
Trevisan [SICOMP'00]). Moreover, the explicit construction of optimal Steiner
trees remains unknown for almost all well-studied high-dimensional point
configurations. Furthermore, a vast majority the state-of-the-art structural
results on (high-dimensional) Euclidean Steiner trees were established in the
1960s, with no noteworthy update in over half a century.
  In this paper, we revisit high-dimensional Euclidean Steiner trees, proving
new structural results. We also establish a link between the computational
hardness of the Euclidean Steiner Tree problem and understanding the optimal
Steiner trees of regular simplices (and simplicial complexes), proposing
several conjectures and showing that some of them suffice to resolve the status
of the inapproximability of the Euclidean Steiner Tree problem. Motivated by
this connection, we investigate optimal Steiner trees of regular simplices,
proving new structural properties of their optimal Steiner trees, revisiting an
old conjecture of Smith [Algorithmica'92] about their optimal topology, and
providing the first explicit, general construction of candidate optimal Steiner
trees for that topology.

Combinatorics;Computational Complexity;Computational Geometry;Metric Geometry