Alexandr Polyanskii: Polynomial plank covering problem

A special case of Bang's plank covering theorem asserts: For any collection of $n$ hyperplanes in the Euclidean space, there is a point in the unit ball at distance at least $1/n$ from the union of the hyperplanes. We are going to discuss the following polynomial generalization of this result (and its relation to the plank covering results of Yufei Zhao and Oscar Ortego-Moreno): For every nonzero polynomial $P\in R[x_1,\dots ,x_d]$ of degree $n$, there is a point of the unit ball in $R^d$ at distance at least $1/n$ from the zero set of the polynomial $P$. Joint work with Alexey Glazyrin and Roman Karasev.