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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!
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    $\begingroup$ What does "separated by the same geodesic distance" mean for two points? $\endgroup$
    – Sridhar Ramesh
    Commented Nov 18, 2012 at 1:25
  • $\begingroup$ @Sridhar: Oh, you are absolutely correct! My mistake, corrected now: I meant separated by a specific distance. Thanks! $\endgroup$
    – Joseph O'Rourke
    Commented Nov 18, 2012 at 1:28
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    $\begingroup$ @JosephO'Rourke: A thread at which you may want to take a look later on: mathoverflow.net/questions/26318/points-on-a-sphere $\endgroup$
    – José Hdz. Stgo.
    Commented Oct 9, 2021 at 21:04

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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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  • $\begingroup$ "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! $\endgroup$
    – Joseph O'Rourke
    Commented Nov 21, 2012 at 1:47
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    $\begingroup$ I think it is the symmetry of the configuration which is relevant... $\endgroup$
    – Igor Rivin
    Commented Nov 21, 2012 at 14:35
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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, assuming that both parameters vary continuously in space.

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    $\begingroup$ If you just want one function, say temperature, it is an amusing exercise with the intermediate value theorem. Nice question for a calculus class. $\endgroup$
    – Michael Murray
    Commented Nov 18, 2012 at 12:44
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    $\begingroup$ 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. $\endgroup$
    – user21349
    Commented Nov 18, 2012 at 16:59
  • $\begingroup$ @Michael Murry: Then you can also restrict to some great circle. There are always two points on the equator with equal temperature. $\endgroup$
    – Eric Naslund
    Commented Nov 18, 2012 at 18:38
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    $\begingroup$ The use of "20 000 km" seems to be a bit less geometrically intuitive than "antipodal". $\endgroup$
    – S. Carnahan ♦
    Commented Nov 19, 2012 at 0:06
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    $\begingroup$ @S. Carnahan: He asked for exact distances in his question. Usually I would say antipodal. $\endgroup$
    – Eric Naslund
    Commented Nov 19, 2012 at 4:16

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