A note on approximation of a ball by polytopes

We study properties of polytopes circumscribed by a unit sphere in Rⁿ with either m extreme points or m facets. We show that if one measures the quality of approximation using the radius of an inscribing sphere then asymptotically the best-possible results are the same for both cases. Somewhat surprisingly, however, the volume can grow substantially faster in m for the case where the polytope has m facets.