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.

MSC:

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

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