On the square peg problem

Abstract

We show that if $\gamma$ is a Jordan curve in ${\mathbb{R}}^2$ which is close to a $C^2$ Jordan curve $\beta$ in ${\mathbb{R}}^2$, then $\gamma$ contains an inscribed square. In particular, if $\kappa >0$ is the maximum unsigned curvature of $\beta$ and there is a map $f$ from the image of $\gamma$ to the image of $\beta$ with $||f(x)-x||<\frac{1}{10\kappa }$ and $f\circ \gamma$ having winding number $1$, then $\gamma$ has an inscribed square of positive sidelength.

11 pages, 3 figures