Random circumscribing polygons and approximations of π

The classical Archimedean approximation of π uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in R2. When n vertices are randomly selected on the circle, a random inscribed polygon can be constructed by connecting adjacent vertices and it is known that its semiperimeter and area both converge to π almost surely as n→∞ and their distributions are also asymptotically Gaussian. In this paper, we consider the case of random circumscribing polygons that are tangent to the circle at each of the prescribed random points. These random versions of the circumscribing Archimedean polygons, however, are more complicated than their inscribed relatives. On the one hand, when all points fall on a semicircle, such a circumscribing polygon does not actually “circumscribe” the circle at all, but instead falls completely outside the circle; on the other hand, even if it behaves normally, its area or semiperimeter can still be unbounded. Nevertheless, we show that such undesirable cases happen with exponentially small probability as n becomes large, and like the case of inscribed polygons, in the limit as n→∞, similar convergence results can be established for the semiperimeters or areas of these random circumscribing polygons.