Central limit theorems for random polygons in an arbitrary convex set

We study the probability distribution of the area and the number of vertices of random polygons in a convex set $K\subset {\mathbb{R}}^2$. The novel aspect of our approach is that it yields uniform estimates for all convex sets $K\subset {\mathbb{R}}^2$ without imposing any regularity conditions on the boundary $\mathrm{\partial }K$. Our main result is a central limit theorem for both the area and the number of vertices, settling a well-known conjecture in the field. We also obtain asymptotic results relating the growth of the expectation and variance of these two functionals.