Compression Theorems and Steiner Ratios on Spheres

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\backslash \bar 3/2$$ \end{document}.