There is Dyson's theorem that states that every continuous real-valued function on the 2-sphere there are four points $p_i$ that form a square around the origin of the sphere and $f(p_1)=f(p_2)=f(p_3)=f(p_4)$.
see: https://www.jstor.org/stable/1969487?origin=crossref
My question is: Can I relax the constraint that the midpoint of the square is the origin of the sphere? More precisely, if I require that the circle around the square is not a great circle of the sphere (as in Dyson's theorem) but a smaller circle of a given radius, is it still possible to find four points forming a (smaller) square such that $f(p_1)=f(p_2)=f(p_3)=f(p_4)$ for a given real-valued function on the sphere?
Thanks in advance for any comments and references! :)
I found a reference with the solution, the answer is yes, a similar statement holds for any rectangle, no matter where the origin lies. The reference is:
https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-62.3.647
see "Theorem D" and the paragraph below "Theorem D"