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Toeplitz' conjecture (also called inscribed square problem) says that:

For every Jordan curve $\mathscr C$, there exists four distincts points $A$, $B$, $C$ and $D$ belonging to $\mathscr C$ such that $ABCD$ is a square.

A Jordan curve is a non self-intersecting continuous loop.

Here is a drawing to illustrate the situation, and a link to the Wikipedia page if you want to find out more about this conjecture.

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The conjecture has already been proven in several cases, including when $\mathscr C$ is piecewise analytic.

So we know that for these two figures, there exists an inscribed square.

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The question is how do I find those squares?

enter image description here

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    Could you describe more explicitly your two polygons? – 2017-02-06
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    @Aretino Not exactly... I was trying to convince myself that Toeplitz' conjecture is indeed true, so I tried it on a few example, but I did not manage to find a square in those particular ones. – 2017-02-06
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    Are you looking for a general algorithm to find a square in any regular curve? – 2017-02-08
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    @Aretino I don't dare to hope that such algorithm would exist (does it?), I was just wondering about that particular example. – 2017-02-08

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