On approximation of a three-dimensional convex body by cylinders
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V. V. Makeev
Translated by: B. M. Bekker - St. Petersburg Math. J. 17 (2006), 315-323
- DOI: https://doi.org/10.1090/S1061-0022-06-00906-X
- Published electronically: February 20, 2006
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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 $\frac {4}{9}(2\sqrt {2}-2)V(K)$.
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Bibliographic Information
- V. V. Makeev
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, Staryĭ Peterhof, St. Petersburg 198904, Russia
- Received by editor(s): May 5, 2004
- Published electronically: February 20, 2006
- © Copyright 2006 American Mathematical Society
- Journal: St. Petersburg Math. J. 17 (2006), 315-323
- MSC (2000): Primary 52B10
- DOI: https://doi.org/10.1090/S1061-0022-06-00906-X
- MathSciNet review: 2159587