Gilbert–Pollak Conjecture on the Euclidean Steiner Ratio

Determine whether the global Steiner ratio of the Euclidean plane R^2 equals sqrt(3)/2 by proving or disproving the Gilbert–Pollak Conjecture, i.e., establish the exact infimum of L(SMT(P))/L(MST(P)) over all finite point sets P subset of R^2.

Background

The paper defines the local and global Steiner ratio and contrasts known results in different geometries. While the hyperbolic plane’s global Steiner ratio is proven to be 1/2, the Euclidean case is governed by the Gilbert–Pollak Conjecture, which asserts that the global Steiner ratio in R^2 equals sqrt(3)/2.

This conjecture directly impacts bounds on how much Steiner minimal trees can reduce total length relative to minimum spanning trees in Euclidean settings, and is cited to contextualize the known hyperbolic upper bound and the comparative difficulty of achieving reductions in Euclidean geometry.