Let $ f $ be a smooth function defined on the sphere such that the set of points where $ f(x) - f(\tilde{x}_y) $ vanishes divides $\mathbb{S}^2$ into exactly four regions for all $y\in \mathbb{S}^2$, where $\tilde{x}_y $ is the reflection of $ x $ with respect to the plane $ P_y $ passing through the origin with normal vector $ y $.

I believe such a function does not exist. My idea is to use Borsuk-Ulam type theorems to prove it. However, I have not been successful in doing so. Any hints would be appreciated.



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