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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

Some geometric properties of closed space curves and convex bodies


Author: V. V. Makeev
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 16 (2004), nomer 5.
Journal: St. Petersburg Math. J. 16 (2005), 815-820
MSC (2000): Primary 51H99
Published electronically: September 23, 2005
MathSciNet review: 2106668
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Abstract: The main results of the paper are as follows.

1. On each smooth closed oriented curve in $ \mathbb{R}^n$, there exist two points the oriented tangents at which form an angle greater than $ \pi/2+\sin^{-1}\frac1{n-1}$.

2. If $ n$ is odd, then an $ (n+1)$-gon with equal sides and lying in a hyperplane can be inscribed in each smooth closed Jordan curve in $ \mathbb{R}^n$. In particular, a rhombus can be inscribed in each closed curve in $ \mathbb{R}^3$.

3. A right prism with rhombic base and an arbitrary ratio of the base edge to the lateral edge can be inscribed in each smooth strictly convex body $ K\subset \mathbb{R}^3$.


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Additional Information

V. V. Makeev
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, Staryĭ Peterhof, St. Petersburg 198904, Russia

DOI: https://doi.org/10.1090/S1061-0022-05-00880-0
Keywords: Convex body, curve, oriented angle, inscribed rhombus, inscribed parallelepiped
Received by editor(s): September 16, 2003
Published electronically: September 23, 2005
Additional Notes: This work was supported by the SS Program (grant no. 1914.2003.1).
Article copyright: © Copyright 2005 American Mathematical Society