Disproofs of generalized Gilbert-Pollak conjecture on the Steiner ratio in three or more dimensions (Q1914000)

The generalized Gilbert-Pollak conjecture states that the ratio \(\ell_{\text{SMT}} (P)/ \ell_{\text{MST}} (P)\), where SMT is the Steiner minimal tree and MST the minimum spanning tree of a point set \(P\), is achieved when \(P\) is the set of vertices of a regular \(d\)-simplex. The authors give an ultra-simplified disproof for \(d \geq 3\) (for \(d = 2\) the conjecture is true) and introduce the concept of ``\(d\)-dimensional'' sausage point sets which seem to have to do with the true minimal ratio point sets.