Assume that $(G,+)$ is an Abelian topological group (maybe locally compact, if necessary) and assume that $V$ is an open connected neighbourhood of zero. Does there exist an open "convex" neighborhood of zer0 $W$ such that $W \subset V$?
A subset $A$ of an Abelian group $G$ we call convex if for each $x\in G$ the condition $2x\in A+A$ implies that $x \in A$. (It is a generalization of notion of convex set in a real linear space).
Thanks.
If $V=G$ then $G$ is such a neighborhood of zero.
If $(G,+)$ is the $\mathbb{Z}/2\mathbb{Z}$ with the discrete topology and $V = \{\operatorname{zero}\}$,
then there is no such neighborhood of zero.
(Since $[1]+[1] = \operatorname{zero} \;$ and $\; [1]\not\in V \:$ .)
Therefore whether or not there is such a neighborhood
of zero depends on things which you did not specify.