Remote Access 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
DOI: https://doi.org/10.1090/S1061-0022-06-00906-X
Published electronically: February 20, 2006
MathSciNet review: 2159587
Full-text PDF Free Access

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?)

  • 1. V. V. Makeev, Affine-inscribed and affine-circumscribed polygons and polyhedra, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 231 (1995), 286-298; English transl., J. Math. Sci. (New York) 91 (1998), no. 6, 3518-3525. MR 1434300 (98b:52004)
  • 2. A. Bielecki and K. Radziszewski, Sur les parallélépipèdes inscrits dans les corps convexes, Ann. Univ. Marie Curie-Sk\lodowska Sect. A 8 (1954), 97-100 (1956). MR 0081498 (18:412g)
  • 3. V. V. Makeev, Inscribed and circumscribed polyhedra of a convex body and related problems, Mat. Zametki 51 (1992), no. 5, 67-71; English transl., Math. Notes 51 (1992), no. 5-6, 469-472. MR 1186533 (93j:52013)
  • 4. -, The Knaster problem and almost spherical sections, Mat. Sb. 180 (1989), no. 3, 424-431; English transl., Math. USSR-Sb. 66 (1990), no. 2, 431-438. MR 0993234 (90d:55005)
  • 5. -, On geometric properties of three-dimensional convex bodies, Algebra i Analiz 14 (2002), no. 5, 96-109; English transl., St. Petersburg Math. J. 14 (2003), no. 5, 781-790. MR 1970335 (2004c:52001)
  • 6. L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum, Grundlehren Math. Wiss., vol. 65, Springer-Verlag, Berlin, 1953. MR 0057566 (15:248b)
  • 7. B. Grunbaum, Affine-regular polygons inscribed in plane convex sets, Riveon Lematematika 13 (1959), 20-24. MR 0108768 (21:7480)
  • 8. V. V. Makeev, Plane sections of convex bodies, and universal fibrations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), 219-233; English transl., J. Math. Sci. (N.Y.) 119 (2004), no. 2, 249-256. MR 1879268 (2002k:52004)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 52B10

Retrieve articles in all journals with MSC (2000): 52B10


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/S1061-0022-06-00906-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

American Mathematical Society