= Closed symmetric absorbing set without an origin neighbourhood
Convexity is essential. In $\mathbb R$, let
$$
S=\{0\}\cup\bigcup_{k\geq0}
\{x:4^{-k}\leq|x|\leq2\cdot4^{-k}\}.
$$
This set is closed, symmetric, and has gaps arbitrarily near zero. Given $x\ne0$, choose $k$ so small-scale that the interval
$[|x|/(2\cdot4^{-k}),|x|/4^{-k}]$ 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.
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