Assume that $f,g:S^{2n}\to \mathbb{R}^{n}$ are $2$ maps. Assume that the set valued map $p(x)=\{f(x),g(x)\}$ is a continuous set valued map. Does there exist a point $p\in S^{2n}$ such that $p(x)=p(-x)$?
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asked Jun 25, 2019 at 14:39
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3$\begingroup$ For $n=1$ this us reduced to usual Borsuk - - Ulam for the map $x\to (f+g,fg)$. $\endgroup$– Fedor PetrovCommented Jun 25, 2019 at 17:11
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$\begingroup$ @FedorPetrov yes. Very intetesting comment. Thanks! $\endgroup$– Ali TaghaviCommented Jun 25, 2019 at 18:28
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1 Answer
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The idea from the comment by Fedor Petrov is suitable for every $n.$ Consider the mapping $F: S^{2n}\to \mathbb{R}^{2n},$ $F(x)=(f(x),g(x)),$ $x\in S^{2n}.$ The Borsuk theorem implies that there exists $x_0\in S^{2n}$ such that $F(x_0)=F(-x_0).$ Obviously, $x_0$ is the desired point.
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$\begingroup$ we do not know $f(x)$ and $g(x)$ separately $\endgroup$ Commented May 9, 2020 at 17:25