Greetings, mathematicians. I know that there’s a theorem that guarantees that there exists a pair of antipodal points on the surface of the Earth, for which the temperature and barometric pressure are equal (assuming that T and P are both continuous scalar functions of 2-D position, of course). This is clearly a higher-dimensional analogue to the Intermediate Value Theorem, and the lower-dimensional problem (one scalar on a circle) is easy to show from the IVT. But I can’t seem to remember what this theorem itself is called, or precisely what it says. This is relevant because I have another problem I’m working on where I’m pretty sure the same theorem would apply, but I don’t know exactly what the theorem states, so I’m not sure.
Possibly the Borsuk-Ulam Theorem?
Definitely.
What can I say, I don’t have a sentimental attachment to this theorem, and I only just now encountered this problem that it’s relevant to. But give me credit: I did remember the theorem itself, just not the names of the dead guys attached to it.
It looks like the Borsuk-Ulam Theorem is more specific than I realized: I don’t actually have an n-sphere, but the interior of a torus, plus some extra conditions. I might be able to construct something using the B-U theorem as a guide, though.
And no, I’d rather not share the exact problem I’m dealing with: This is for a contest, and I’d feel guilty getting too much help for it.
Oh, and as an aside: You were right in that old thread. It may be possible to use the IVT to rigorously prove the B-U Theorem in an intuitive way, but it’s at the very least much harder than it looks.
I’ve also already tried using IVT-based techniques on the problem I’m currently working on, but I’ve already found that singularities can crop up in my arguments, and that they can be (and possibly must be) at exactly the most inconvenient point for my purposes.