On the square peg problem

TL;DR

This paper proves that curves close to a smooth Jordan curve in the plane necessarily contain an inscribed square, extending the classical square peg problem to near-smooth curves with specific curvature and mapping conditions.

Contribution

It establishes conditions under which a Jordan curve close to a $C^2$ curve inscribes a square, broadening the class of curves known to have inscribed squares.

Findings

Curves close to a $C^2$ Jordan curve contain an inscribed square.

A quantitative condition involving curvature and a mapping ensures inscribed squares.

The result applies to curves with bounded curvature and specific proximity to smooth curves.

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.