Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratios
Cole Hugelmeyer · arXiv (Cornell University) · 2019
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.