Let $M$ be a closed additive submonoid of $\mathbb{R}^n$ with $n\geq1$. Suppose also that there exists $r>0$ such that every ball of radius $r$ intersects $M$. I wonder if we can obtain more information on $M$. For example, if $n=1$, it is easy to see that $M$ has to be a subgroup of $\mathbb{R}$, thanks to the classical characterization of such subgroups. Is it still true when $n\geq2$?