Let $n ≥ 2$ and $g : \mathbb{S}^n \to \mathbb{R}^2$ be a map such that $g(−x) = −g(x)$ for all $x$. Prove that there is a point $x_0$ in $\mathbb{S}^ n$ such that $g(x_0) = 0.$
How to proceed?
2026-04-01 13:04:18.1775048658
Anitpodal Map for Sphere, Mean Value
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Hint: $S^n$ contains $S^2$. Restrict $g$ to this $S^2$, where it is still the case $g(-x)=-g(x)$.