Limit theorems for areas and perimeters of random inscribed and circumscribed polygons

We find the limiting distributions for the maximal area of random convex inscribed polygons and for minimal area of random convex circumscribed polygons whose vertices are distributed on the circumference with almost arbitrary continuous density. These distributions belong to the Weibull family. From this we deduce new limit theorems in the case when the vertices of polygons have the uniform distribution on the ellipse. Some similar theorems are formulated also for perimeters of inscribed and circumscribed polygons.