Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratios

TL;DR

This paper proves that every smooth Jordan curve inscribes rectangles with aspect ratios covering at least one third of all possible ratios, using topological and geometric methods involving projective planes and torus knots.

Contribution

It establishes a lower bound of one third for the measure of inscribed rectangle aspect ratios in smooth Jordan curves, combining topology, geometry, and knot theory.

Findings

At least one third of aspect ratios are inscribed in any smooth Jordan curve.

Sets of disjoint projective planes can be totally ordered.

A sharp lower bound of 1/3 is proven for intersection probabilities involving torus knots.

Abstract

We prove that for every smooth Jordan curve $\gamma$, if $X$ is the set of all $r \in [0,1]$ so that there is an inscribed rectangle in $\gamma$ of aspect ratio $\tan(r\cdot \pi/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedded in $\mathbb{R}\times \mathbb{R}P^3$. We prove that any such set of projective planes can be equipped with a natural total ordering. We then combine this total ordering with Kemperman's theorem in $S^1$ to prove that $1/3$ is a sharp lower bound on the probability that a M\"obius strip filling the $(2,1)$-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.