Could you think of any usages of the closest pair of points and the farthest pair of points in any dimension and any metric?
Here are my ideas.
The closest pair of points:
- testing - we have a set of points which are e.g. some laboratories owned by some company; the company has build a teleporter; it can be reconstructed only in these labs because e.g. their walls were build with gold and it's necessary for a teleporter to be encapsulated in gold room for some reason; but a teleporter produces a magnetic field and if they build 2 teleporters in one faculty the fields will interfere and they won't work; and ofc they want to test them and it would be good to first test them for as small distance as possible; so they have to find two labs which are closest to each other
- many travels - points are hospitals; each hospital has exactly 1kg of penicillin (or any medicine); Mr. X needs 1.5kg of penicillin each day to live; each hospital get this 1kg of drug every day; therefore, X has to visit exactly 2 hospitals every day; he'll move to any part of the country but he doesn't want to travel every day very far
- genetics - points are people; we want to find two people the has the closest DNA (it doesn't concert us how to calculate the distance between two people); why? because e.g. they want to find out how close we can be to each other and what is the least number of nucleobases we have to change in a person to make him sufficiently different
- genetics II - points are people but we measure the distance between them in looks; e.g. we are close to annihilation and we need to send a person on a mission where he will likely die; and we have a device necessary to accomplish the mission but it works for only one person which it recognises by looks (and it's verification mechanism is very precise) so we need to find two as similar people as possible
The farthest pair of points:
- testing - we've successfully tested the teleported for the smallest distance, the deadline is near so we want to check it as thoroughly as possible; so we need to find two laboratories that are as far from each other as possible
- escape / safety - postapocalyptic world; only some places (points) are habitable; people can live on one of them as they need to keep close to each other for safety (there are few of them); but living in some place can cause a chemical cloud (lethal) which will expand to unknown sizes; and it may happen only once for some reason; so they have to live in such a place for which they can find as farthest other habitat as possible
- genetics - we want to find two organisms as different as possible in order to experiment on them and get even more diverse species or just to check how diverse e.g. mammals can be
I know these are "a litte" far-fetched but the important usages are before the dash (testing, genetics, etc.). Instead of teleporters there could be something else, more probable to exist but the usage is the same. There might be other practical uses for finding two as similar organisms are possible but I'd have to ask a geneticist. But feel free to write more practical stories to my usages.
It would be superb to have an example for bigger dimensions.
Applications of closest pair.
The following algorithms and applications can be implemented efficiently using closest pair data structures, or involve closest pair computation as important subroutines.
Source
As for your ideas:
Applications of farthest pair.
About your ideas, again they don't seem wrong, but as you say a little far-fetched.
Well, the applications I mentioned above can go to higher dimensions, e.g. the convex hull. Another example can come from the Bioinformatics field, where you have many molecules and you want to find the closest pair for some biological reason.