Typical faces of best approximating three-polytopes

Károly J. Böröczky, Péter Tick and Gergely Wintsche

Abstract: 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), 271--320.

Keywords: polytopal approximation, extremal problems

Classification (MSC2000): 52A27, 52A40

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