2020-10-18 A nice exercise in metric spaces

It’s been a very long time since I posted anything math-y here (more than five years!). It certainly has something to do with the fact that I no longer work as a mathematician (even if I am part of one maths-related project, but that’s a secret). A few days ago, however, after a short discussion with a friend, I came up with a really nice exercise. Note: it is not difficult at all, quite the contrary, but I like it a lot.

Hopefully everyone reading this knows what a metric space is (and if you don’t, no offence, but this exercise is probably not meant for you). There is one interesting class of metric spaces, called the “ultrametric spaces”, where instead of the usual triangle inequality we assume that d(x,y)≤max{d(x,z),d(y,z)} for every x,y,z in the space in question. (Sorry for un-LaTeX-y formatting here, I just noticed that LaTeX fragment rendering on my blog is broken, and I don’t have time at the moment to investigate it – and we’ll need very few equations anyway.) This is a much stronger condition, and it has some rather strange consequences (like every point belonging to a ball being its center). One of the unusual properties of an ultrametric space is that “every triangle is isosceles”, that is, every triple of points has one point equidistant (“equidistant” – so grown-up!) from the other two.

And here is the exercise, coming in two parts. First of all: do there exist any metric spaces with the same property but not being ultrametric? In other words, must a metric space in which every triangle is isosceles be ultrametric? And the other question is to characterize all metric spaces where every triangle is equilateral, that is, d(x,y)=d(y,z)=d(z,x) for every x,y,z.

As I said, these exercises are trivial for any seasoned mathematician, but might be a good thing to ask students. And even though I do find them very easy, I really like them from aesthetic point of view. And – finally – I am not bound by some stupid obligations like publishing a novel (or “novel”) paper once or twice per year – so I can have fun with little toy problems like that whenever I feel like it.

Happy solving!

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