×

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
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Boy, Werner, \"{U}ber die {C}urvatura integra und die {T}opologie geschlossener {F}l\"{a}chen, Math. Ann.. Mathematische Annalen, 57, 151-184 (1903) · JFM 34.0537.07
[2] Hugelmeyer, C., Every smooth {J}ordan curve has an inscribed rectangle of aspect ratio \(\sqrt{3} (2018)\)
[3] Kemperman, J. H. B., On products of sets in a locally compact group, Fund. Math.. Polska Akademia Nauk. Fundamenta Mathematicae, 56, 51-68 (1964) · Zbl 0125.28901
[4] Kirby, R., What is ...{B}oy’s surface?, Notices Amer. Math. Soc.. Notices of the American Mathematical Society. https://www.ams.org/journals/notices/200710/200710FullIssue.pdf, 54, 1306-1307 (2007) · Zbl 1151.53306
[5] Matschke, Benjamin, A survey on the square peg problem, Notices Amer. Math. Soc.. Notices of the American Mathematical Society, 61, 346-352 (2014) · Zbl 1338.51017
[6] Raikov, D., On the addition of point-sets in the sense of {S}chnirelmann, Rec. Math. [Mat. Sbornik] N.S., 5(47), 425-440 (1939) · Zbl 0022.21003
[7] \v{S}nirel \('\) man, L. G., On certain geometrical properties of closed curves, Uspehi Matem. Nauk. Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 10, 34-44 (1944)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.