On 9/25/13, Dan Asimov <dasimov@earthlink.net> wrote:
Yeah, I wondered the same thing. My guess is, It's true. But that ain't a proof.
On 2013-09-25, at 12:58 PM, Fred Lunnon wrote:
Another generalisation which occurs to me concerns the maximum content of an arbitrary simplex inscribed in the hypercube. Might the maximal simplex be regular? I don't know the answer, even for 2-space.
Also, Fred, those are very nice maximal regular simplex results. Can you please say -- or repeat if I missed it -- how your optimizer program works?
--Dan
No great imaginative leap to reveal, I'm afraid. The search program just plonks a little simplex at the centre of the hypercube, then repeatedly applies a random rotation, followed by translating to touch the hypercube boundary, and dilating to touch the opposite half --- assuming that the entire transformation increases the simplex diameter. The rotation angle is bounded; after a specified number of consecutive failures to improve, the angle bound is reduced by a specified ratio. Then I have to tinker with input values and trawl through the results, on the lookout for an exceptionally large side or (approximate) symmetry. If I get lucky, I replace the clusters of numerically close coordinate components by variables, then maximise the resulting side analytically, subject to the regularity constraint. So far, the answer has always come out satisfyingly just a little larger still than the largest experimental value. The Monte-Carlo search stage is of course excruciatingly slow to converge: partly on account of the high dimension (n+1)n/2 ; and partly because it is iterating towards a critical value, an inherently ill-conditioned procedure. Fred Lunnon