  First, we study constructible subsets of $\A^n_k$ which contain a line in any
direction. We classify the smallest such subsets in $\A^3$ of the type
$R\cup\{g\neq 0\},$ where $g\in k[x_1,...,x_n]$ is irreducible of degree $d$,
and $R\subset V(g)$ is closed. Next, we study subvarieties $X\subset\A^N$ 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.
