# Square Peg Problem and curve density

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?
• 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. – Aleksei Kulikov Jun 26 '18 at 10:28