Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 48, No. 2, pp. 521545 (2007) 

Typical faces of best approximating threepolytopesKároly J. Böröczky, Péter Tick and Gergely WintscheAlfréd Rényi Institute of Mathematics, Budapest, PO Box 127, H1364, Hungary, email: carlos@renyi.hu and Department of Geometry, Roland Eötvös University, Budapest, Pázmány Péter sétány 1/C, H1117, Hungary; Budapest, Gy\H ur\H u utca 24., H1039, Hungary, email: tick@renyi.hu; Institute of Mathematics, Roland Eötvös University, Budapest, Pázmány Péter sétány 1/C, H1117, Hungary, email: wgerg@ludens.elte.huAbstract: For a given convex body $K$ in $\R^3$ with $C^2$ boundary, let $P_n^i$ be an inscribed polytope of maximal volume with at most $n$ vertices, and let $P_{(n)}^c$ be a circumscribed polytope of minimal volume with at most $n$ faces. P. M. Gruber [G] proved that the typical faces of $P_{(n)}^c$ are asymptotically close to regular hexagons in a suitable sense if the Gau{ß}Kronecker curvature is positive on $\partial K$. In this paper we extend this result to the case if there is no restriction on the Gau{ß}Kronecker curvature, moreover we prove that the typical faces of $P_n^i$ are asymptotically close to regular triangles in a suitable sense. In addition writing $P_{(n)}$ and $P_n$ to denote the polytopes with at most $n$ faces or $n$ vertices, respectively, that minimize the symmetric difference metric from $K$, we prove the analogous statements about $P_{(n)}$ and $P_n$. [G] Gruber, P. M.: Optimal configurations of finite sets in Riemannian $2$manifolds. Geom. Dedicata {\bf 84} (2001), 271320. Keywords: polytopal approximation, extremal problems Classification (MSC2000): 52A27, 52A40 Full text of the article:
Electronic version published on: 7 Sep 2007. This page was last modified: 28 Jun 2010.
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