On Mon, 29 Sep 2003, Dan Hoey wrote:
At any rate, I would like to mention a modification to the approach I outlined on Friday. In this approach we place points in k-cells of side epsilon that intersect the given k-1-sphere. I will assign a point a "nominal location" of the center of its cell. However, since the point may differ from that location by epsilon.sqrt(k)/2, we must enforce a separation of points by r + epsilon.sqrt(k). In addition, since we require that each point be at the minimum distance to at least two neighbors, we count those points within r - epsilon.sqrt(k) of the given point toward the count of required near neighbors.
There's something about the logic of Allan's and Dan's proposals that really bugs me, namely that they have their quantifiers the wrong way round. Namely, the purpose of Jud's program was to establish upper bounds, because that's where the problem is. Let me make use of my own personal hotline to the truth, and say straight out that I "know" the answer to be 24. Then the only problem is to prove that fact to the mathematical community who don't trust my hotline. It's trivial to prove that it's at least 24 - I just show them the 24-sphere configuration. So all that remains is to prove 25 impossible. But look at Dan's words above: they are addressed to making sure that a solution exists, not to proving that it doesn't. The same was true of Allan's, and makes them both irrelevant.
I'm curious about how this approach might work (or how it may have worked already, since it's a fairly standard approach to geometrical searching). I remark that it not could not only tell us which (n,k,r) are not feasible, but it could provide witnesses for which (n,k,r-epsilon.sqrt(k)) are feasible.
It's obvious that it won't work as stated, at least to establish 24 as an upper bound, because it's addressed to the lower bound problem. As regards lower bounds, it would obviously be far less efficient than the "Gosset-type" approach, because it's still doing the "universalist" searching that would be appropriate to finding upper bounds. Anyway, we don't need better lower bounds, because 24, which has been known for more than a century, is actually best possible. Now if you changed some of the signs and inequalities around, so that it was properly addressed to upper bounds, it suffers from a very big defect, namely that those epsilons make it necessarily not "exact". What this means is, that if there were a 25-sphere example, but only one that required exact placement of the spheres, it couldn't possibly rule it out, and so couldn't possibly do the only thing we want it to. I felt this way on first reading that stuff about "the TC"; namely that the feeling that this idea could rescue something of the argument was just upside down (to use a kind phrase). All it could possibly do was provide an "upper bound" that wouldn't in fact be an upper bound, because it wasn't provable. The number 7 has that property, dammit! You folks need to get your logic straight! Regards, JHC