Branched Coverings and Steiner Ratio

TL;DR

This paper proves that in branched locally isometric coverings, the Steiner ratio of the base space is at least that of the total space, with applications to specific geometric surfaces showing they share the Euclidean plane's Steiner ratio.

Contribution

It establishes a comparison theorem for Steiner ratios in branched coverings and applies it to surfaces like tetrahedra and cones, linking their ratios to the Euclidean plane.

Findings

Steiner ratio of the surface of an isosceles tetrahedron equals that of the Euclidean plane.

Steiner ratio of a flat cone with angle 2π/k equals that of the Euclidean plane.

The Steiner ratio of the base is not less than that of the covering space.

Abstract

For a branched locally isometric covering of metric spaces with intrinsic metrics, it is proved that the Steiner ratio of the base is not less than the Steiner ratio of the total space of the covering. As applications, it is shown that the Steiner ratio of the surface of an isosceles tetrahedron is equal to the Steiner ratio of the Euclidean plane, and that the Steiner ratio of a flat cone with angle of $2\pi/k$ at its vertex is also equal to the Steiner ratio of the Euclidean plane.