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The following analogue of Kakutani's theorem has been proved by F. J. Dyson (MR44620)

Theorem (F. J. Dyson 1951): Let $\Bbb S^2$ be the surface of a sphere with center $Z$ in Euclidean $3$-space $\Bbb R^3$, and let $f(x)$ be a continuous real-valued function defined on $\Bbb S^2$. Then there exist four points $x_1 , x_2 , x_3 , x_4$ on $\Bbb S^2$ forming the vertices of a square with center $Z$, such that $$f(x_1)=f(x_2)=f(x_3)=f(x_4).$$

Is the above theorem true for $\Bbb R^2=\Bbb S^2\backslash\{\infty\}$? i.e. are there four points $x_1 , x_2 , x_3 , x_4$ on $\Bbb R^2$ forming the vertices of a square with center $Z$ (probably the origin), such that $$f(x_1)=f(x_2)=f(x_3)=f(x_4)?$$

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No, let $f(x,y)=x$. Then the level sets of $f$ are straight lines $x=c$, and a straight line cannot contain a square.

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  • $\begingroup$ Does this works for any plan? we know that $\Bbb R^2\subset\Bbb R^3$. if we consider $yz$-plan then I think this does not work? $\endgroup$
    – C.F.G
    Commented Dec 5, 2017 at 16:51

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