Generalized Gilbert–Pollak Steiner ratio conjecture

Establish the validity of the generalized Gilbert–Pollak conjecture on the Steiner ratio in Euclidean spaces of dimension three and higher, which posits a specific optimal ratio between Steiner tree and minimum spanning tree lengths.

Background

The paper resolves a special case (“Simplex is the Best for Graph Embeddings”) relevant to high-dimensional Euclidean Steiner Tree hardness, but notes that this result is a special case of the much broader generalized Gilbert–Pollak conjecture, which remains open.

Confirming or refuting the generalized conjecture would have significant implications for the geometry of Steiner trees and approximation hardness in high-dimensional settings.

References

The simplest such conjecture, called the Simplex is the Best for Graph Embeddings Conjecture in , was a special case of the widely open generalized Gilbert-Pollak conjecture .

— Accelerating Scientific Research with Gemini: Case Studies and Common Techniques  (2602.03837 - Woodruff et al., 3 Feb 2026) in Computational Geometry: Steiner Trees, Section 4.2 (Research Context)