Geodesic
On polygons inscribed in a closed space curve
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 11, Tome 372 (2009), pp. 97-102 
V. V. Makeev. On polygons inscribed in a closed space curve. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 11, Tome 372 (2009), pp. 97-102. http://geodesic.mathdoc.fr/item/ZNSL_2009_372_a8/
@article{ZNSL_2009_372_a8,
     author = {V. V. Makeev},
     title = {On polygons inscribed in a~closed space curve},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {97--102},
     year = {2009},
     volume = {372},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_372_a8/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $n$ be an odd positive integer. It is proved that if $n+2$ is a power of a prime number and $\gamma$ is a regular closed non-self-intersecting curve in $\mathbb R^n$, then $\gamma$ contains vertices of an equilateral $(n+2)$-link polyline with $n+1$ vertices lying in a hyperplane. It is also proved that if $\gamma$ is a rectifiable closed curve in $\mathbb R^n$, then $\gamma$ contains $n+1$ points that lie in a hyperplane and divide $\gamma$ into parts one of which is twice as long as each of the others. Bibl. – 5 titles.

[1] L. G. Shnirelman, “O nekotorykh geometricheskikh svoistvakh zamknutykh krivykh”, UMN, 1944, no. 10, 34–44 | MR | Zbl

[2] H. Griffiths, “The topology of square pegs in round holes”, Proc. London Math. Soc. (3), 62 (1991), 647–672 | DOI | MR | Zbl

[3] V. V. Makeev, “O nekotorykh geometricheskikh svoistvakh zamknutykh prostranstvennykh krivykh i vypuklykh tel”, Algebra i analiz, 16:5 (2004), 92–100 | MR | Zbl

[4] V. V. Makeev, “O chetyrëkhugolnikakh, vpisannykh v zamknutuyu krivuyu, i eë vershinakh”, Zap. nauchn. semin. POMI, 299, 2003, 241–251 | MR

[5] V. V. Makeev, “O chetyrëkhugolnikakh, vpisannykh v zamknutuyu krivuyu”, Mat. zametki, 57:1 (1995), 129–132 | MR | Zbl