Statistical approximation of plane convex sets

Summary

Given a convex set F in the plane with a sufficiently smooth boundary we try to approximate it by polygons in the following way. Using some specified sampling procedure we pick out n points on the boundary. Through each such point we draw the tangent. Consider the polygon F*_n spanned by all these tangents. If n is large we would expect F*_n to be close to F. Measuring the deviation by the area of F*_n — F we will derive an asymptotic expression for this area when n becomes large. This expression can be used to choose the optimum sampling procedure in the sense of smallest asymptotic deviation.

The problem arose from a problem of statistical approximation in propositional calculus, see section 1.