John Conway <conway@math.princeton.edu> wrote:
The feeling that I'd expressed too strongly my criticisms of Jud McCranie's proposed attack on the kissing number problem somehow convinced me that they were wrong, and led me to apologise....
You know, I had been about to do something like that, too. I think it was all the "wolg'ing that was so irritating. A little "wolg" goes a long way, and after two "wolg"s and an "etc" I was about to "wolg" all over my keyboard. But looking closer, I think there's something correct in Jud's idea, perhaps new or perhaps not. It might even be possible to use it to (correctly) prove bounds on the r-kissing number (how many radius-r balls can kiss a unit ball in k dimensions) for which I'm sure there are plenty of open cases. First, let's try it without using symmetry or requiring any adjacencies of the balls (except, of course, to the central one). Quantitize the space of ball centers into small pieces p1,p2,...,pm. This could be done as coordinate-wise, so that each pi is the intersection of a dimension-k box with the (1+r)-sphere, or perhaps there is a better way of chopping up the space. A pair of pieces (pi,pj) is admissible if they contain respective points (xi,xj) at a distance of at least 2r. If that's too hard to test, admissiblity standards can be loosened. For instance, we could call (pi,pj) admissible if the underlying boxes contain an appropriate pair of points. We then exhaustively search for a set of n pieces that are pairwise admissible. If there is no such set, then the (r,k) kissing number is less than n. Of course, this doesn't prove the converse. But I think that if the kissing number is less than n, then there is a minimum overlap epsilon that will occur in any configuration of n spheres. If we require the pieces (or their boxes, in the loosened regime) to have dimension at most 2 epsilon, then the search for a pairwise adminssible subset will fail. This can be improved for efficiency. If n >= k I think that we can require each ball to kiss k-1 others in addition to the central one. If that's incorrect, we can at least require some adjacencies. This will allow us to require that adjacent balls be mapped to pieces that contain respective points (xi,xj) at a distance of exactly 2r. Other reductions of the search space are possible from symmetry. I imagine this method will be intractible in the interesting cases, but at least it's correct. Dan P.S. Thanks Hartmut and JHC for reminding me that conjugacy induces a homomorphism! --D[o]H