It is well known that convex implies connected, and it is clear that if $X$ is a locally convex topological vector space and $\emptyset \neq A\subseteq X$ is open then $A$ contains a nonempty open convex set.
Question. Does there exist a topological vector space $X$ and a nonempty open connected set $A\subseteq X$ such that $A$ does not contain any nonempty open convex subset?
Yes. For instance, consider the space $X=L^p([0,1])$ for some $p\in (0,1)$, and take $A:=\{x\in X\mid \lVert x\rVert_p<1 \}$.
Then $A$ is open and path-connected (for every $a\in A$, $t\mapsto ta$ yields a path from $0$ to $a$), but it does not contain any nonempty convex open subset (the only convex open sets in $X$ are $\emptyset$ and $X$).