Abstract
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 ${\mathrm{tan}}(r\cdot \pi /4)$, 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.
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@ARTICLE{Boy,
author = {Boy, Werner},
title = {Über die {C}urvatura integra und die {T}opologie geschlossener {F}lächen},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {57},
year = {1903},
number = {2},
pages = {151--184},
issn = {0025-5831},
mrclass = {DML},
mrnumber = {1511204},
doi = {10.1007/BF01444342},
url = {https://doi.org/10.1007/BF01444342},
jfmnumber = {34.0537.07},
} -
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@MISC{Me,
author= {Hugelmeyer, C.},
title = {Every smooth {J}ordan curve has an inscribed rectangle of aspect ratio $\sqrt{3}$},
arxiv = {1803.07417},
year = {2018},
zblnumber = {},
} -
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@ARTICLE{Kemperman,
author = {Kemperman, J. H. B.},
title = {On products of sets in a locally compact group},
journal = {Fund. Math.},
fjournal = {Polska Akademia Nauk. Fundamenta Mathematicae},
volume = {56},
year = {1964},
pages = {51--68},
issn = {0016-2736},
mrclass = {22.20},
mrnumber = {0202913},
mrreviewer = {A. B. Paalman-de Miranda},
doi = {10.4064/fm-56-1-51-68},
url = {https://doi.org/10.4064/fm-56-1-51-68},
zblnumber = {0125.28901},
} -
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@ARTICLE{Kirby,
author = {Kirby, R.},
title = {What is ...{B}oy's surface?},
journal = {Notices Amer. Math. Soc.},
fjournal = {Notices of the American Mathematical Society},
volume = {54},
number = {10},
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@ARTICLE{survey,
author = {Matschke, Benjamin},
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journal = {Notices Amer. Math. Soc.},
fjournal = {Notices of the American Mathematical Society},
volume = {61},
year = {2014},
number = {4},
pages = {346--352},
issn = {0002-9920},
mrclass = {51M04},
mrnumber = {3184501},
mrreviewer = {Zdenka Kolar-Begović},
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url = {https://doi.org/10.1090/noti1100},
zblnumber = {1338.51017},
} -
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@ARTICLE{Raikov,
author = {Raikov, D.},
title = {On the addition of point-sets in the sense of {S}chnirelmann},
journal = {Rec. Math. [Mat. Sbornik] N.S.},
volume = {5(47)},
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mrclass = {10.0X},
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url = {http://mi.mathnet.ru/eng/msb/v47/i2/p425},
zblnumber = {0022.21003},
} -
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@ARTICLE{Sch,
author = {Šnirel$'$man, L. G.},
title = {On certain geometrical properties of closed curves},
journal = {Uspehi Matem. Nauk},
fjournal = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
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issn = {0042-1316},
mrclass = {56.0X},
mrnumber = {0012531},
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zblnumber = {0060.35107},
}