Maximum vs Minimum Distance Between a Random Set of Points

I am using center of mass calculations to find the minimum distance between a random set of points on a plane. Can the maximum distance between the same points be calculated that is the minimum the same way or is there another technique. And, could the minimum distance point be related to the maximum distance point that is minimum. Please let me know what you think. Thank you.

[Sven]

Can you show us what you've done? It's a little unclear from the description what you mean by "can the maximum distance between the same points be calculated that is the minimum the same way"...

Answer 1

[Walter Roberson]

No, the quantities cannot be related that way.

Suppose you put a number of points in a tight clump, such as along the edge of a small circle. Now suppose you put one other point a large distance away.

Consider the point 'A' on the circle that is closest to the outlier point 'X'. 'A' is very close to all points except the outlier X, so for that point on the circle, 'A', the maximum distance to any other point will be the distance to the outlier X. Likewise, for all the other points on the circle, the maximum distance to the other points would be the distance to the outlier X. As we defined 'A' as being the closest point on the circle to the outlier X, the minimum maximum distance will be A<->X .

Is that minimum maximum distance the same as the maximum distance? No. The other points on the circle are all further away from X and so have greater maximum distances. The point(s) closest to the projection of XA through the circle will, in this situation, be the one(s) with the maximum distance.

Is the minimum maximum distance going to involve one of the points that is involved with the maximum distance? That I do not know. My suspicion is that it might not, but that is only a suspicion and I have not thought about enough cases yet to have any certainty in it.