Abstract
Suppose AiBiCi (i = 1, 2) are two triangles of equal side lengths lying on spheres Φi with radii r1, r2 (r1 < r2) respectively. First we prove the existence of a map h: A1B1C1 → A2B2C2 so that for any two points P1, Q1 in A1B1C1,¦P1Q1¦≥¦h(P1)h(Q1)¦. Moreover, if P1, Q1 are not on the same side, then the inequality strictly holds. This compression theorem can be applied to compare the minimum of a variable in triangles on two spheres. Hence, one of the applications of the compression theorem is the study of Steiner minimal tress on spheres. The Steiner ratio is the largest lower bound for the ratio of the lengths of Steiner minimal trees to minimal spanning trees for point sets in a metric space. Using the compression theorem we prove that the Steiner ratio on spheres is the same as on the Euclidean plane, namely \(\backslash \bar 3/2\).
Similar content being viewed by others
References
Du, D. Z., and F. K. Hwang. (1992). A proof of the Gilbert-Pollak Conjecture on the Steiner ratio, Algorithmica, 7, 121–135.
Gilbert, E. N., and H. O. Pollak. (1968). Steiner minimal trees, SIAM J. Appl. Math., 16, 1–29.
Hwang, F. H. (1976). On Steiner minimal trees with rectilinear distance, SIAM J. Appl. Math., 30, 104–114.
Karp, R. M. Reducibility among combinatorial problems, Complexity of Computer Computation (Edited by R. E. Miller and J. W. Tatcher), Plenum Press, New York, 1972, 85–103.
Rubinstein, J. H., and D. A. Thomas. (1991). A variational approach to the Steiner network problem, Ann. Oper. Res., 33, 481–499.
Todhunter, I., and J. G. Leathem. Spherical trigonometry, Macmillan and CO., London, 1901.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rubinstein, J.H., Weng, J.F. Compression Theorems and Steiner Ratios on Spheres. Journal of Combinatorial Optimization 1, 67–78 (1997). https://doi.org/10.1023/A:1009711003807
Issue Date:
DOI: https://doi.org/10.1023/A:1009711003807