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St. Petersburg Mathematical Journal

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Inscribed and circumscribed polyhedra for a convex body and continuous functions on a sphere in Euclidean space


Author: V. V. Makeev
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 18 (2006), nomer 6.
Journal: St. Petersburg Math. J. 18 (2007), 997-1009
MSC (2000): Primary 52A10, 52A15
DOI: https://doi.org/10.1090/S1061-0022-07-00979-X
Published electronically: October 2, 2007
MathSciNet review: 2307358
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Abstract | References | Similar Articles | Additional Information

Abstract: Two related problems concerning continuous functions on a sphere $ S^{n-1} \subset {\mathbb{R}}^n$ are studied, together with the problem of finding a family of polyhedra in $ {\mathbb{R}}^n$ one of which is inscribed in (respectively, circumscribed about) a given smooth convex body in $ {\mathbb{R}}^n$. In particular, it is proved that, in every convex body $ K\subset{\mathbb{R}}^3$, one can inscribe an eight-vertex polyhedron obtained by ``equiaugmentation'' of a similarity image of any given tetrahedron of class $ T$.


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Additional Information

V. V. Makeev
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospect 28, Staryĭ Peterhof, St. Petersburg 198904, Russia
Email: mvv57@inbox.ru

DOI: https://doi.org/10.1090/S1061-0022-07-00979-X
Keywords: Convex body, inscribed and circumscribed polyhedra
Received by editor(s): May 20, 2005
Published electronically: October 2, 2007
Article copyright: © Copyright 2007 American Mathematical Society

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