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

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.