Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratios. (English) Zbl 1472.51011

Summary: 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 Möbius strip filling the \((2,1)\)-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.


51M05 Euclidean geometries (general) and generalizations
51G05 Ordered geometries (ordered incidence structures, etc.)
28A12 Contents, measures, outer measures, capacities
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