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arXiv · 2312.09552

The number of inscribed and circumscribed graphs of a convex polyhedron

Yagub N. Aliyev
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Abstract

In the paper we prove that the number of graphs inscribed into graph of a convex polyhedron and circumscribed around another graph does not exceed 4. For this we first studied Poncelet type problem about the number of convex $n$-gons inscribed into one convex $n$-gon and circumscribed around another convex $n$-gon. It is proved that their number is also at most 4. This contrasts with Poncelet type porisms where usually infinitude of such polygons is proved, provided that one such polygon already exists. An inequality involving ratio of lengths of line segments is used. Alternative way of using Maclaurin-Braikenridge's conic generation method is also discussed. Properties related to constructibility with straightedge and compass are also studied. A new proof, based on mathematical induction, of generalized Maclaurin- Braikenridge's theorem is given. We also gave examples of regular polygons and a polyhedron for which number 4 is realized.

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Yagub N. Aliyev. 2023-12-15. The number of inscribed and circumscribed graphs of a convex polyhedron. https://doi.org/10.1007/s00454-025-00775-w

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