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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

On approximation of a three-dimensional convex body by cylinders


Author: V. V. Makeev
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 17 (2005), nomer 2.
Journal: St. Petersburg Math. J. 17 (2006), 315-323
MSC (2000): Primary 52B10
Published electronically: February 20, 2006
MathSciNet review: 2159587
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Abstract | References | Similar Articles | Additional Information

Abstract: New results on approximation of a convex body $ K\subset \mathbb{R}^3$ by affine images of circular cylinders, parallelepipeds, hexagonal and octagonal regular (and some other) prisms are obtained.

Two of the theorems obtained are as follows ($ V(K)$ denotes the volume of a body $ K\subset \mathbb{R}^3$).

Theorem 1. Let $ K$ be an arbitrary convex body in $ \mathbb{R}^3$. There exists a regular octagonal prism an affine image of which is circumscribed about $ K$ and has volume at most $ 3\sqrt{2}V(K)$, and there exists a circular cylinder an affine image of which is circumscribed about $ K$ and has volume at most $ \frac{3\pi}{2}V(K)$. For a tetrahedron $ K$ both estimates are the best possible.

Theorem 2. Let $ K$ be a centrally symmetric convex body in $ \mathbb{R}^3$. There exists a regular octagonal prism, an affine image of which lies in $ K$ and has volume at least $ \frac{4}{9}(2\sqrt{2}-2)V(K)$.


References [Enhancements On Off] (What's this?)

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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: http://dx.doi.org/10.1090/S1061-0022-06-00906-X
PII: S 1061-0022(06)00906-X
Keywords: Volume, cylinder, prism, parallepiped
Received by editor(s): May 5, 2004
Published electronically: February 20, 2006
Article copyright: © Copyright 2006 American Mathematical Society