Let $X$ be a topological vector space. Let us call a set $S\subset X$ almost-convex if, for every $x,y\in S$ and every $V$ a neighborhood of zero, there exists a continuous path from $x$ to $y$ entirely contained in $([x,y]+V)\cap S$, where $[x,y]$ denotes the segment with end-points $x,y$. (In other words the path in $S$ is close to the segment.)
How does this notion relate to (local) pathwise connectedness?
I can prove that if $S$ is almost-convex then $S$ is pathwise connected and locally pathwise connected while $\overline{S}$(the closure of $S$) is convex and, clearly, every convex set is almost-convex.
I am interested in a converse of the form: if $S$ is locally pathwise connected with $\overline{S}$ convex then $S$ is almost-convex.