The Inscribed Square Problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A Jordan curve in the plane is the image $\Gamma=\gamma(S^1)$ of a continuous injective map $\gamma:S^1\to\mathbb R^2$, where $S^1=\{(x,y):x^2+y^2=1\}$. Let $R(x,y)=(-y,x)$ denote rotation through a right angle. An inscribed square means four distinct points of $\Gamma$ that are the vertices of a nondegenerate Euclidean square; its sides need not lie on the curve.

Statement recorded in the supplied source.

For every Jordan curve $\Gamma\subseteq\mathbb R^2$, are there $c\in\mathbb R^2$ and $u\in\mathbb R^2\setminus\{0\}$ such that \[ c+u,\quad c+Ru,\quad c-u,\quad c-Ru \quad\text{all belong to }\Gamma? \] No smoothness, rectifiability, Lipschitz or convexity hypothesis is imposed on the Jordan curve. [S2]

Short English statement

Must every simple closed continuous plane curve contain the four vertices of a square of positive side length?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1163 (GEO-008). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Joshua Evan Greene and Andrew Lobb, Square pegs between two graphs, arXiv:2407.07798 (2024), Section 1, definition of inscription and the Square Peg Problem. https://arxiv.org/abs/2407.07798.