  We establish central limit theorems for natural volumes of random inscribed
polytopes in projective Riemannian or Finsler geometries. In addition, normal
approximation of dual volumes and the mean width of random polyhedral sets are
obtained. We deduce these results by proving a general central limit theorem
for the weighted volume of the convex hull of random points chosen from the
boundary of a smooth convex body according to a positive and continuous density
in Euclidean space. In the background are geometric estimates for weighted
surface bodies and Berry-Esseen bounds for functionals of independent random
variables.
