Edge lengths guaranteed to form a simplex

Simplexes in spaces of constant curvature

“…In spite of these different-looking examples there is a uniform criterion -very useful in theoretical considerations -for determining the realizability of an arbitrary allowable matrix. The Euclidean case of this Smatrix criterion was given a short proof in [1] and that short proof is extended here to cover the hyperbolic and spherical cases.…”

“…For hyperbolic geometry such a model is furnished by the Poincar6 ball and for spherical geometry, by the stereographic image of the unit sphere in ~" ÷ 1 THEOREM 1. Let L = (lu) be an (n + 1) x (n + 1) allowable matrix and let its entries be used to form the n x n matrix S = (sij) where t cosh li,+ cosh l~,+-cosh I u in the hyperbolic case 1 …”

“…Otherwise we can assume that 2,+1(/) ~< m = minpipj < 1 i#j and the condition m < l 0 < I on the off-diagonal entries of an (n + 2) x (n + 2) allowable matrix L places L in the interior of the appropriate sup norm ball and guarantees that L has an (n + 1)-dimensional realization. We complete the proof of the main result by generalizing some Euclidean lemmas from [ 1] to the setting X". These lemmas show that C is one of the other configurations described in Section 2 and hence m --2, + 1(/).…”

“…If some distance satisfies m < llj < l then it is possible to move the points of C within X" to make them all satisfy this inequality. (See [1] for details.) This is a contradiction because the resulting points would then form a non-degenerate (n + 1)-simplex.…”

“…Proof If ab = m we use Lemma 4 and the method of [1] to replace every other edge-length m in C with a variable edge-length x. We increase x from m to 1 stopping at Xo > m when ab(xo) = l and thereby produce a degenerate (n + 1)-simplex whose edge-lengths are l and Xo > m. If Xo < l, this is a contradiction because such an (n+ 1)-simplex must be non-degenerate.…”