On Friday 24 October 2008, Dan Asimov wrote: [ratio of maximal to minimal distance given n points in the plane]
(I think I have an asymptotic formula for the optimal ratio as n -> oo, but so far not a proof. I'm curious if anyone else comes up with a conjectural asymptotic formula that is asymptotic to mine.
Putative asymptotic formula is way, way below.)
[spoiler space preserved -- gjm]
Guess: r(n) ~ sqrt(3n/2) as n -> oo.
Naively, the best we can do is going to be something like hexagonal packing in a circle. Let's give the circle radius 1, so area pi, so our points all live in hexagons of area pi/n. In the obvious packing of hexagons with centres d apart, the area of each hexagon is sqrt(3/4) d^2, so d = sqrt(pi/n / sqrt(3/4)). On the other hand, the max distance is going to be ~ 2. So the ratio is 2 / (sqrt(pi*sqrt(3)/2) / sqrt(n)) = sqrt(n) sqrt(sqrt(3)/2pi), modulo any stupid algebraic errors I've made. -- g