Approximation of convex bodies by axially symmetric bodies

Let $C$ be an arbitrary planar convex body. We prove that $C$contains an axially symmetric convex body of area at least $\frac{2}{3}\vert C\vert$. Also approximation by some specific axially symmetric bodies is considered. In particular, we can inscribe a rhombus of area at least $\frac{1}{2}\vert C\vert$ in $C$, and we can circumscribe a homothetic rhombus of area at most $2\vert C\vert$ about $C$. The homothety ratio is at most $2$. Those factors $\frac{1}{2}$ and $2$, as well as the ratio $2$, cannot be improved.