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At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?

What is the strongest, most impressive statement one can make here? The Borsuk-Ulam Theorem applies, but I am uncertain of its full implications. Could one say that the two points are (1) separated by a specific geodesic distance, (2) have the same temperature, and (3) have the same barometric pressure? For example...? I pose this question for its pedagocial import, but it clearly follows from known theorems.

To what extent do these results extend to $\mathbb{R}^d$ for $d>3$? Thank you for your help!
            (Wikipedia image)

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What does "separated by the same geodesic distance" mean for two points? –  Sridhar Ramesh Nov 18 '12 at 1:25
    
@Sridhar: Oh, you are absolutely correct! My mistake, corrected now: I meant separated by a specific distance. Thanks! –  Joseph O'Rourke Nov 18 '12 at 1:28

2 Answers 2

up vote 9 down vote accepted

One of the standard generalizations is Knaster's conjecture: for every function $f: \mathbb{S}^{n-1}\rightarrow \mathbb{R}^m, m\lt n,$ and $k=n-m+1$ points $p_1, \dots, p_k \in \mathbb{S}^{n-1}$ does there always exista rotation $\rho \in SO(n),$ such that $f(\rho(p_1) = \dots = f(\rho(p_k)).$ That this is true for $k=2$ is a theorem of H. Hopf (which generalizes Borsuk-Ulam). It turns out that Knaster's conjecture is true for some $m, n$ and false for others. See this nice paper by Hinrich and Richter for more results and references.

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"In particular, for the central case of real-valued functions $f$, i.e., $m = 1$, $k = n$, H. Yamabe and Z. Yujobo confirmed the conjecture if $\lbrace p_1, \ldots, p_n \rbrace$ is an orthonormal basis." Interesting that the geometry of the points is needed for this conclusion. Thanks, Igor! –  Joseph O'Rourke Nov 21 '12 at 1:47
    
I think it is the symmetry of the configuration which is relevant... –  Igor Rivin Nov 21 '12 at 14:35

We can say the following:

"At any given time, there are two points on the earth exactly 20 000 km apart with the same exact same temperature and barometric pressure."

I am making a few assumptions, but do note that the distance from the north pole to the south pole is 20 000 km. Indeed, in the Wikipedia article which you linked to in your question, we find the quote:

"The case n = 2 is often illustrated by saying that at any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures. This assumes that temperature and barometric pressure vary continuously."

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@Eric: Thanks for the quote! –  Joseph O'Rourke Nov 18 '12 at 2:19
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If you just want one function, say temperature, it is an amusing exercise with the intermediate value theorem. Nice question for a calculus class. –  Michael Murray Nov 18 '12 at 12:44
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The result loses some of its mystery and impressiveness if you just count degrees of freedom. You get to choose two numbers, latitude and longitude. You're trying to satisfy two constraints, equal temperature and equal pressure. –  Ben Crowell Nov 18 '12 at 16:59
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The use of "20 000 km" seems to be a bit less geometrically intuitive than "antipodal". –  S. Carnahan Nov 19 '12 at 0:06
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@S. Carnahan: He asked for exact distances in his question. Usually I would say antipodal. –  Eric Naslund Nov 19 '12 at 4:16

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