TITLE: There are two points on the Earth's surface that ... ?
QUESTION [15 upvotes]: 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)

REPLY [14 votes]: 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.