In Research Problems in Discrete Geometry by Peter Brass, William O. J. Moser, János Pach, in section 2.1 page 75, "Decomposition of multiple packings and coverings", problems with a similar flavour are listed, some with solutions, about covering of the plane with discs, or the space with spheres. Typically, for spheres, for some $a$ and $b$, $\{a\le F_x\le b, \forall x\}$ entails that there exists a partition $F=\tilde F\cup \hat F$ with $\{\tilde F_x\ge 1, \forall x\}$ and the same for $\hat F$. A probabilistic proof uses the local Lovasz lemma.

In Research Problems in Discrete Geometry by Peter Brass, William O. J. Moser, János Pach, in section 2.1 page 75, "Decomposition of multiple packings and coverings", problems with a similar flavour are listed, some with solutions, about covering of the plane with discs, or the space with spheres. Typically, for a covering $F$ of the space by unit spheres, such that for some $a$ (large enough), $$\{a\le F_x\le 2^{\tfrac{a}3-8}, \forall x\}$$ there always exists a partition $F=\tilde F\cup \hat F$ with $\{\tilde F_x\ge 1, \forall x\}$ and $\{\hat F_x\ge 1, \forall x\}$. A probabilistic proof uses the local Lovasz lemma.