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.