Geodesic
Polygons inscribed in a closed curve and a three-dimensional convex body
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 116-125 Cet article a éte moissonné depuis la source Math-Net.Ru

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Here are samples of results obtained in the paper. Let $\gamma$ be a centrally symmetric closed curve in $\mathbb R^n$ that does not contain its center of symmetry, $O$. Then $\gamma$ is circumscribed about a square (with center $O$), and about a rhombus (also with center $O$) whose vertices split $\gamma$ into parts of equal length. If $n$ is odd, then there is a centrally symmetric equilateral $2n$-link polyline inscribed in $\gamma$ and lying in a hyperplane. Let $K\subset\mathbb R^3$ be a convex body, $x\in(0;1)$. Then $K$ is circumscribed about an affine-regular pentagonal prism $P$ such that the ratio of the lateral edge $l$ of $P$ to the longest chord of $K$ parallel to $l$ is equal to $x$. Bibl. – 7 titles. 
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V. V. Makeev. Polygons inscribed in a closed curve and a three-dimensional convex body. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 116-125. http://geodesic.mathdoc.fr/item/ZNSL_2008_353_a9/

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