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

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 $\tan (r\cdot \pi \mathrm{/}4)$, then the Lebesgue measure of $X$ is at least $1\mathrm{/}3$. 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 $1\mathrm{/}3$ 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.