Let the N+1 vertices of an N-simplex be the columns of an Nx(N+1) matrix. We can demand this matrix be upper triangular by translating the simplex so first column is (0,0,0,...,0) and then using the R (upper triangular) factor in the QR factorization of the remaining NxN matrix (Q is orthogonal). If we then deleted the first row, then we would get an (N-1)-dimensional simplex, which generically (i.e. if that first row had all entries unequal) would have every edge length shorter than it used to be before the deletion. If this (N-1)-simplex fits inside the original N-dimensional simplex, then that proves the original N-simplex was Rupertable. Unfortunately for that argument's importance, shortening edge lengths does not necessarily imply "fits inside." For example, a triangle with sides 999, 999, 9 cannot fit inside an equilateral triangle with sides 1000. Nevertheless, for many simplices I suspect some variant of that approach will succeed in Ruperting it. I would like to establish that all simplices are Rupertable, or find a counterexample. And if we have two N-simplices and want to know whether one fits inside another, that might be a hard problem because there are (N+1)! different ways to try. This "can it fit inside" problem seems at least as hard as the graph isomorphism problem. Specifically if I had a "fit inside" oracle for simplices then I could use it to solve all GI problems trivially as follows: The graphs must have the same number of vertices, edges and same valency-tuples in order to be isomorphic (of course) so say "no" if not. Let two simplex vertices A,B have distance 1+1000^(-N) if graph has edge AB, otherwise distance is 1. Ask oracle whether simplex #1 fits inside simplex #2 scaled by 1+1000000^(-N). "Yes" <==> graphs isomorphic. Laszlo Babai has recently claimed in an 84-page "preliminary" paper, to have a proof that graph isomorphism is soluble in "quasipolynomial" time, meaning exp(k*(logN)^c) time for some positive constants c and k: http://arxiv.org/abs/1512.03547v1 If correct, Babai's result is a major advance in computer science, one of the greatest in years. Before Babai the best runtime guarantee known was N^O(sqrt(N*logN)). -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)