We propose a variant of Cauchy's Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogov's Comparison Theorem, Legendre's Theorem and Cauchy's Arm Lemma.
Abel, Zachary; Charlton, David; Collette, Sébastien; Demaine, Erik D.; Demaine, Martin L.; Langerman, Stefan; O'Rourke, Joseph; Pinciu, Val; and Toussaint, Godfried, "Cauchy’s Arm Lemma on a Growing Sphere" (2008). Computer Science: Faculty Publications. 42.