On an extremal problem for polynomials in $n$ variables

This article is devoted to an examination of the following extremal problem: find the quantity
$$ C_{k,n}(\lambda,B)=\sup_{|\omega|\ge\lambda}\sup_{P\in\mathscr P_{k,n}(\omega)}\|P\|_{C(B)}, $$
where $B$ is an $n$-dimensional sphere and $\mathscr P_{k,n}(\omega)$ is the totality of polynomials $P$ of degree $k$ in $n$ variables for which $\|P\|_{C(\omega)}\le1$. Here $\omega$ is a measurable set from $B$ and the first sup is taken over all measurable $\omega\subset B$ having measure $|\omega|\ge\lambda$.
The exact order of growth of $C_{k,n}(\lambda, B)$ which respect to $\lambda$ as $\lambda\to0$ is found in this article. Various applications of the results are examined as well.