  We consider a minimizing variant of the well-known \emph{No-Three-In-Line
Problem}, the \emph{Geometric Dominating Set Problem}: What is the smallest
number of points in an $n\times n$~grid such that every grid point lies on a
common line with two of the points in the set? We show a lower bound of
$\Omega(n^{2/3})$ points and provide a constructive upper bound of size $2
\lceil n/2 \rceil$. If the points of the dominating sets are required to be in
general position we provide optimal solutions for grids of size up to $12
\times 12$. For arbitrary $n$ the currently best upper bound for points in
general position remains the obvious $2n$. Finally, we discuss the problem on
the discrete torus where we prove an upper bound of $O((n \log n)^{1/2})$. For
$n$ even or a multiple of 3, we can even show a constant upper bound of 4. We
also mention a number of open questions and some further variations of the
problem.
