 Closed symmetric absorbing set without an origin neighbourhood

Convexity is essential. In $\mathbb{R}$, letThis set is closed, symmetric, and has gaps arbitrarily near zero. Given $x\ne 0$, choose $k$ so small-scale that the interval
$[|x|/(2\cdot 4^{-k}),|x|/4^{-k}]$ has length at least one; it contains an integer $n$, and then $x/n\in S$. Thus ${\bigcup }_{n}nS=\mathbb{R}$, although $S$ is not a neighbourhood of zero.