  We consider the online hitting set problem for the range space $\Sigma=(\cal
X,\cal R)$, where the point set $\cal X$ is known beforehand, but the set $\cal
R$ of geometric objects is not known in advance. Here, objects from $\cal R$
arrive one by one. The objective of the problem is to maintain a hitting set of
the minimum cardinality by taking irrevocable decisions. In this paper, we
consider the problem when objects are unit balls or unit hypercubes in
$\mathbb{R}^d$, and the points from $\mathbb{Z}^d$ are used for hitting them.
First, we address the case when objects are unit intervals in $\mathbb{R}$ and
present an optimal deterministic algorithm with a competitive ratio of~$2$.
Then, we consider the case when objects are unit balls. For hitting unit balls
in $\mathbb{R}^2$ and $\mathbb{R}^3$, we present $4$ and $14$-competitive
deterministic algorithms, respectively. On the other hand, for hitting unit
balls in $\mathbb{R}^d$, we propose an $O(d^4)$-competitive deterministic
algorithm, and we demonstrate that}, for $d<4$, the competitive ratio of any
deterministic algorithm is at least $d+1$. In the end, we explore the case
where objects are unit hypercubes. For hitting unit hypercubes in
$\mathbb{R}^2$ and $\mathbb{R}^3$, we obtain $4$ and $8$-competitive
deterministic algorithms, respectively. For hitting unit hypercubes in
$\mathbb{R}^d$ ($d\geq 3$), we present an $O(d^2)$-competitive randomized
algorithm. Furthermore, we prove that the competitive ratio of any
deterministic algorithm for the problem is at least $d+1$ for any
$d\in\mathbb{N}$.
