  We consider an online version of the geometric minimum hitting set problem
that can be described as a game between an adversary and an algorithm. For some
integers $d$ and $N$, let $P$ be the set of points in $(0, N)^d$ with integral
coordinates, and let $\mathcal{O}$ be a family of subsets of $P$, called
objects. Both $P$ and $\mathcal{O}$ are known in advance by the algorithm and
by the adversary. Then, the adversary gives some objects one by one, and the
algorithm has to maintain a valid hitting set for these objects using points
from $P$, with an immediate and irrevocable decision. We measure the
performance of the algorithm by its competitive ratio, that is the ratio
between the number of points used by the algorithm and the offline minimum
hitting set for the sub-sequence of objects chosen by the adversary.
  We present a simple deterministic online algorithm with competitive ratio
$((4\alpha+1)^{2d}\log N)$ when objects correspond to a family of $\alpha$-fat
objects. Informally, $\alpha$-fatness measures how cube-like is an object. We
show that no algorithm can achieve a better ratio when $\alpha$ and $d$ are
fixed constants. In particular, our algorithm works for two-dimensional disks
and $d$-cubes which answers two open questions from related previous papers in
the special case where the set of points corresponds to all the points of
integral coordinates with a fixed $d$-cube.
