Variants of the Kakeya problem over an algebraically closed field

First, we study constructible subsets of ${\text{A}}_k^n$ which contain a line in any direction. We classify the smallest such subsets in ${\text{A}}^3$ of the type $R\cup \{g\mathrm{\ne }0\},$ where $g\in k[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ is irreducible of degree $d$, and $R\subset V(g)$ is closed. Next, we study subvarieties $X\subset {\text{A}}^N$ for which the set of directions of lines contined in $X$ has the maximal possible dimension. These are variants of the Kakeya problem in an algebraic geometry context.