Largest rectangle in a convex polygon

The main result of Marek Lassak, "Approximation of convex bodies by rectangles", Geom. Dedicata 47 (1993), 111–117, doi:10.1007/BF01263495 is:

Let $C$ be a convex body in the plane. We can inscribe a rectangle $R$ in $C$ such that a homothetic copy $S$ of $R$ is circumscribed about $C$. The positive homothety ratio is at most 2 and $\frac12|S|\le|C|\le 2|R|$.

($|\cdot|$ denotes area.) In particular, $k\le 2$, which is optimal as noted in comments. According to Lassak, the fact that every convex body contains a rectangle $R$ with $|C|\le 2|R|$ was shown in K. Radziszewski, "Sur une problème extrémal relatif aux figures inscrites et circonscrites aux figures convexes", Ann. Univ. Mariae Curie-Sklodowska, Sect. A 6 (1952), 5–18.