Jordan curves inscribe a positive measure of rectangles
Joshua Evan Greene, Andrew Lobb

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 is a Jordan curve of diameter which encloses a region of area . We prove that there exists a subset of measure at least such that if , then there exist four points on at the vertices of a rectangle whose diagonals meet at angle .
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