It has been proved by Radziszewski in this paper

K. Radziszewski. Sur une probleme extremal relatif aux gures inscrites et circonscrites aux gures convexes. Ann. Univ. Mariae Curie-Sklodowska, Sect. A6, pages 5-18, 1952.

that the area of the largest inscribed rectangle (LIR) inside a convex polygon (C) is at least one half of the area of C, i.e. Area(LIR)>=Area(C)/2.

Unfortunately the paper is very old and I couldn't find it. But I think the proof should not be very long. Does anyone have access to this paper or have any suggestion about how to prove it?