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

Hugelmeyer, Cole

Ann. Math. (2) 194, No. 2, 497-508 (2021).

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

Keywords:

inscribed rectangles; Jordan curve; aspect ratio

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) · Zbl 0060.35107