Adam Goucher writes: Instead I propose the following way to compare packings:
For a discrete configuration of points C, we define f_C(R) to be the minimum, amongst all balls of radius R, of the number of points in C which lie within this ball.
Then, we say that C dominates D (written C >= D) if there exists some R_0 such that for all R >= R_0, we have f_C(R) >= f_D(R).
I like this, but I think the sharpness of the cutoff at distance R creates difficulties. For instance, try comparing two hexagonal close packings of points in the plane using this definition. My approach to avoiding these difficulties is a smoothed-out version of Adam's proposal. You give each point a weight that's close to 1 when the point is near the origin and goes to 0 as the point goes to infinity, with R replaced by a distance parameter that measures the spread. This is my integration/summation kernel. Of course it needs to be integrable/summable. This does not give us a partial order, but it does give us a sense in which all the hexagonal close packings appear to "beat" all the other point-packings in which no inter-point distances lie in (0,1). Jim