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By Károly (h-eotvo-g) Böröczky

Abstract

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 www.renyi.hu/∼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 www.renyi.hu/∼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 www.renyi.hu/∼carlos/radiusasymp.pdf

Year: 2011
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