# Branched coverings and Steiner ratio

@article{Ivanov2016BranchedCA, title={Branched coverings and Steiner ratio}, author={Alexander Ivanov and Alexey A. Tuzhilin}, journal={Int. Trans. Oper. Res.}, year={2016}, volume={23}, pages={875-882} }

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 at its vertex is also equal to the Steiner ratio of the Euclidean plane.

## References

SHOWING 1-10 OF 21 REFERENCES

### Steiner Ratio for Manifolds

- Mathematics
- 2003

The Steiner ratio characterizes the greatest possible deviation of the length of a minimal spanning tree from the length of the minimal Steiner tree. In this paper, estimates of the Steiner ratio on…

### The Steiner and Gromov–Steiner Ratios and Steiner Subratio in the Space of Compacta in the Euclidean Plane with Hausdorff Distance

- Mathematics
- 2014

The Steiner and Gromov–Steiner ratios and Steiner subratio are fundamental characteristics of metric spaces. In this work, an attempt is made to find these ratios for the space of compacta in…

### A continuity criterion for Steiner-type ratios in the Gromov-Hausdorff space

- Mathematics
- 2014

A criterion for the continuity of the Steiner ratio, the Steiner subratio, and the Steiner-Gromov ratio in the space of all compact metric spaces with the Gromov-Hausdorff metric is obtained. It is…

### Steiner Ratio for Hadamard Surfaces of Curvature at Most k < 0

- Mathematics
- 2014

In this paper, the Hadamard surfaces of curvature at most k are investigated, which are a particular case of Alexandrov surfaces. It was shown that the total angle at the points of an Hadamard…

### Some NP-complete geometric problems

- MathematicsSTOC '76
- 1976

We show that the STEINER TREE problem and TRAVELING SALESMAN problem for points in the plane are NP-complete when distances are measured either by the rectilinear (Manhattan) metric or by a natural…

### A proof of the Gilbert-Pollak conjecture on the Steiner ratio

- MathematicsAlgorithmica
- 2005

This paper provides a proof for Gilbert and Pollak's conjecture that for any P, Ls(P)≥(√3/2)Lm(P), and denotes the lengths of the Steiner minimum tree and the minimum spanning tree on P.

### On the Steiner Ratio in

- MathematicsDiscret. Math. Algorithms Appl.
- 2011

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization and the best previously known lower bound for the Steiner ratio in the literature is 0.

### Steiner ratio for hyperbolic surfaces

- Mathematics
- 2006

: We prove that the Steiner ratio for hyperbolic surfaces is 1 / 2.

### One-dimensional Gromov minimal filling problem

- Mathematics
- 2012

The paper is devoted to a new branch in the theory of one-dimensional variational problems with branching extremals, the investigation of one-dimensional minimal fillings introduced by the authors.…