# Set valued version of Borsuk Ulam theorem

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 $$p(x)=p(-x)$$?

• For $n=1$ this us reduced to usual Borsuk - - Ulam for the map $x\to (f+g,fg)$. – Fedor Petrov Jun 25 at 17:11
• @FedorPetrov yes. Very intetesting comment. Thanks! – Ali Taghavi Jun 25 at 18:28