Skip to main content
Article thumbnail
Location of Repository

Previous Up Next Article Citations From References: 1 From Reviews: 1

By Károly (h-eotvo-g) Böröczky


Polytopes of minimal volume with respect to a shell—another characterization of the octahedron and the icosahedron. (English summary) Discrete Comput. Geom. 38 (2007), no. 2, 231–241. The authors consider convex bodies in E 3 which contain a unit ball. Let r> 1 be given. The extremal points of the bodies are of distance at least r from the centre of that ball. The authors show that the optimal bodies for r = √ 3 are regular octahedra and for r = √ 15 − 6 √ 5 are regular icosahedra. These optimal bodies have minimal volume and minimal surface. A conjecture is that the optimal bodies for r = 3 are regular tetrahedra. For the proof of these results the authors use a lemma which says that the optimal facets are regular triangles. Furthermore, one can see that the analogues of the presented assertion do not hold for the cube and for the dodecahedron. Reviewed by J. Böhm References 1. K. Bezdek: On a stronger form of Roger’s lemma and the minimum surface area of Voronoi cells in unit ball packings. J. Reine Angew. Math., 518 (2000), 131–142. MR1739407 (2001b:52033) 2. K. Böröczky, K.J. Böröczky: A stability property of the octahedron and the icosahedron. Submitted. Available at∼carlos/radiusstab.pdf. 3. K. Böröczky, K.J. Böröczky, G. Wintsche: Typical faces of extremal polytopes with respect to a thin threedimensional shell. Period. Math Hungar., accepted. Available at∼carlos/radiustyp.pdf. 4. K. Böröczky, K.J. Böröczky, C. Schütt, G. Wintsche: Convex bodies of minimal volume, surface area and mean width with respect to thin shells. Canad. J. Math., accepted. Available at∼carlos/radiusasymp.pdf

Year: 2011
OAI identifier: oai:CiteSeerX.psu:
Provided by: CiteSeerX
Download PDF:
Sorry, we are unable to provide the full text but you may find it at the following location(s):
  • (external link)
  • (external link)
  • (external link)
  • (external link)
  • Suggested articles

    To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.