I got a little bit lucky for n = 9 , guessing a configuration which plonks one simplex vertex just outside the hypercube --- but scaled down to fit, still gives lower bound 9*(2*Sqrt(5) - 3*Sqrt(2)) = 2.065457... on the maximal side length --- but that is unlikely to be sharp. The geometric problem I formulated earlier doesn't appear amenable to conventional "quadratic programming" or "convex programming" techniques, which apparently cope only with quadratic inequalities. It can be formulated in a nutshell: given the values of some subset comprising more than half the coordinate components of the vertices of a regular simplex, complete its coordinates. Fred Lunnon On 9/30/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Still contemplating Dan's n-space regular simplices inscribed in a unit hypercube, the way one does, for n = 9 now.
Given that for n = 7,8 the (conjectural) max side equals 2 , it was not a promising start that my rotation-jiggling search program took 12 hours to find just a couple of examples exceeding that: but the good-ish news is that for n = 9 a lower bound on the maximal side is 2.00452263 .
Such configurations as warrent further investigation do not seem to have a great deal evident in the way of symmetry. More promising is their preponderance --- in one case two-thirds --- of coordinate components near --- within say 0.02 --- of an integer --- 0 or 1 --- which, one may assume, are on the way to eventually converge to interval endpoints.
The freedom 1 + (n+1)n/2 similarity transforming current into limiting configuration is therefore by this stage well over-determined. If only there were an effective algorithm to compute it ...
I ran up program which forces these near-endpoints to endpoints, in the process, destroying the regularity of the simplex, then attempts to restore regularity by minimising the side-length variance, with respect to individual cooordinate components iteratively. This works a treat on smaller cases already solved, but disappointingly fails to progress on the problem to hand.
A less direct approach --- solving for the almost orthogonal projective matrix of the final transformation --- involves large numbers of simultaneous quadratic equations. This looks fairly horrible --- "quadratic programming", perhaps?
Fred Lunnon