Compression theorems and Steiner ratios on spheres. (English) Zbl 0895.90173

Rubinstein, J. H.; Weng, J. F.

J. Comb. Optim. 1, No. 1, 67-78 (1997).

Summary: Suppose \(A_iB_iC_i\) \((i=1,2)\) are two triangles of equal side lengths lying on spheres \(\Phi_i\) with radii \(r_1,r_2\) \((r_1< r_2)\) respectively. First we prove the existence of a map \(h: A_1B_1C_1\to A_2B_2C_2\) so that for any two points \(P_1,Q_1\) in \(A_1B_1C_1\), \(| P_1Q_1|\geq | h(P_1)h(Q_1)|\). Moreover, if \(P_1,Q_1\) 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 trees 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 \(\sqrt{3}/2\).

MSC:

90C35

Programming involving graphs or networks

Keywords:

spheres; compression theorem; Steiner minimal trees on spheres; Steiner ratio; minimal spanning trees