Random circumscribing polygons and approximations of ππ

The classical Archimedean approximation of ππ uses the semiperimeter or area of regular   polygons inscribed in or circumscribed about a unit circle in R2R2. When nn 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 ππ almost surely as n→∞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 nn becomes large, and like the case of inscribed polygons, in the limit as n→∞n→∞, similar convergence results can be established for the semiperimeters or areas of these random circumscribing polygons.