Balanced but not convex? In a topological vector space $X$, a subset $S$ is convex if \begin{equation}tS+(1-t)S\subset S\end{equation} for all $t\in (0,1)$. 
$S$ is balanced if \begin{equation}\alpha S\subset S\end{equation} for all $|\alpha|\le 1$.
So if $S$ is balanced then $0\in S$, $S$ is uniform in all directions and $S$ contains the line segment connecting 0 to another point in $S$. 
Due to the last condition it seems to me that balanced sets are convex. However I cannot prove this, and there are also evidence suggesting the opposite.
I wonder whether there is an example of a set that is balanced but not convex.
Thanks!

The interior of a regular pentagram centered at the origin is balanced but not convex.
