Polygons inscribed in a~closed curve and a~three-dimensional convex body

V. V. Makeev

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 10, Tome 353 (2008), pp. 116-125

Résumé

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.