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

Central limit theorems for random polygons in an arbitrary convex set