There are two points on the Earth's surface that ... ? 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)
 A: 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."
A: 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.
