Jordan curves inscribe a positive measure of rectangles

Published: April 18, 2026PDF arXiv

Abstract

Suppose that $γ\subset \mathbb{C}$ is a Jordan curve of diameter $2R$ which encloses a region of area $A$. We prove that there exists a subset $I \subset (0,π)$ of measure at least $A/R^2$ such that if $θ\in I$, then there exist four points on $γ$ at the vertices of a rectangle whose diagonals meet at angle $θ$.

Jordan curves inscribe a positive measure of rectangles

Joshua Evan Greene, Andrew Lobb

math.GT math.CO math.SG

Published: April 18, 2026PDF arXiv

Abstract

Suppose that $γ\subset \mathbb{C}$ is a Jordan curve of diameter $2R$ which encloses a region of area $A$. We prove that there exists a subset $I \subset (0,π)$ of measure at least $A/R^2$ such that if $θ\in I$, then there exist four points on $γ$ at the vertices of a rectangle whose diagonals meet at angle $θ$.