Asymptotic normality for random polytopes in non-Euclidean geometries

Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and central limit theorems in Lutwak’s dual Brunn–Minkowski theory are established. The results follow from a central limit theorem for weighted random polytopes in Euclidean spaces. In the background are Stein’s method for normal approximation and geometric properties of weighted floating bodies.