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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?

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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).

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  • $\begingroup$ What's the title of the review by W Gustin? $\endgroup$ – MB_MB Nov 13 '13 at 3:20
  • $\begingroup$ Sorry, it's Gustin's review of Radziszewski's paper in Math Reviews. MR0054268 (14, 896f), as noted. $\endgroup$ – Gerry Myerson Nov 13 '13 at 3:23
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Lassak improved Radziszewski's result in Approximation of convex bodies by rectangles as follows:

Theorem 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 $|S|/2\le |C| \le 2|R|$.

The paper is written in English and relatively new.

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  • $\begingroup$ I am aware of that paper but I need to see Radziszewski's proof. $\endgroup$ – MB_MB Nov 13 '13 at 3:21

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