Jordan curves inscribe a positive measure of rectangles

TL;DR

The paper proves that for a Jordan curve of certain size and enclosed area, there is a positive measure set of angles for which rectangles with vertices on the curve exist, with diagonals meeting at those angles.

Contribution

It establishes a lower bound on the measure of angles for which rectangles inscribed in a Jordan curve exist, linking geometric properties to measure theory.

Findings

Existence of rectangles with vertices on the curve for a positive measure set of angles.

Lower bound of A/R^2 on the measure of angles with inscribed rectangles.

Rectangles' diagonals meet at angles within this measure set.

Abstract

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