Let $u$ be a smooth function defined on the sphere $\mathbb{S}^2$, and let $R \in \mathrm{SO}(3)$ be a three-dimensional rotation. Define $$ S_R = \{x \in \mathbb{S}^2 : u(x) \neq u(Rx)\}. $$
Suppose there exists a constant $\alpha > 0$ (independent of $R$) such that the area of each connected component of $S_R$ is at least $\alpha$, for all rotations $R$ with $S_R \neq \emptyset$. Does this imply any symmetry property about the function $u$? Must there exist $R\neq I$ such that $S_R =\emptyset$?
