Hi, all. Is this a known theorem, meaning you've seen it stated somewhere? A simple quadrilateral whose minimum distance between vertices is 1 and whose maximum distance is sqrt(2) is a unit square.
The statement is mine but the idea is not. Someone else is analyzing an algorithm for finding the closest pair out of a set of points. That person wants to prove an equivalent statement: the maximum number of points that can be placed in a unit square is 4 when the minimum distance between points is 1.
It seems obvious but developing a proof has not been. I have one now, but I did not want to bother posting it if this is, say, a well-known exercise from high-school geometry.