4 problems

Let $S^{d-1}\subset\mathbb R^d$ be the unit sphere, and let zones be centrally symmetric parts of the sphere covered by planks. Fejes Tóth's conjecture. If finitely many zones cove…

Polynomial plank conjecture. There exists a point $p\in B^d$ such that, for every $k=1,\ldots,N$, the distance from $p$ to the zero set of $P_k$ is at least $\delta_k$.

Smurov's conjecture. The planks can be translated so that their translates cover $B$.

Smurov's conjecture. There exists a constant $C_d$, possibly depending on $d$, such that any collection of planks in $\mathbb R^d$ with total width at least $C_d$ can be translated…