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?



Largest inscribed rectangle inside a convex polygon
The result also appeared in a shorter paper, Constantin Radziszewski, Sur un problème extrémal relatif aux figures inscrites dans les figures convexes, C. R. Acad. Sci. Paris 235 (1952) 771–773, MR0054268 (14,896f). The review by W Gustin summarizes the proof. 


There is also a proof in Wilhelm Süss, Ueber Parallelogramme und Rechtecke, die sich ebenen Eibereichen einbeschreiben lassen, Rend. Mat. e Appl. (5) 14 (1955) 338–341, MR0069523 (16,1046b). 


