Abstract. Let $K$ be a convex body in ${\msbm E}^3$ with a $C^2$ smooth boundary. In this article, we investigate polytopes with at most $n$ edges circumscribed about $K$ or inscribed in $K$, which approximate $K$ best in the Hausdorff metric. The asymptotic behaviour of the distance, as a function of $n$, of such best approximating polytopes and $K$ is known, see [3] for an asymptotic formula. In this article, we prove that the typical faces of the best approximating circumscribed or inscribed polytopes in the Hausdorff metric with at most $n$ edges are asymptotically squares with respect to the second fundamental form of $\partial K$.

AMS Subject Classification (1991): 52A27, 52A50

Keyword(s): polytopal approximation, extremal problems, Hausdorff distance

Received March 6, 2008, and in revised form May 22, 2008. (Registered under 1/2008.)