Square Peg Problem (or conjecture) is so famous. See this article

Let $CS:=\{\gamma:S^1\longmapsto\mathbb{R}^2 | \;\;\text {Square Peg Problem is true}\}$ and $C=\{\Upsilon:S^1\longmapsto\mathbb{R}^2 \}$.

Is this true that $CS$ is dense in the set $C$? Is there any results in this direction?

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    $\begingroup$ Well, wikipedia says that polygons are in $CS$ and if they are not dense in $C$ in topology you are thinking about then you should definitely tell us which topology you are imposing on $C$. Actually, for dense subset of polygons it is obvious that they are in $CS$ -- start with arbitrary polygon and add small square bump on one of it sides. $\endgroup$ – Aleksei Kulikov Jun 26 '18 at 10:28

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