For smooth surfaces in the euclidean space there are notions of the mean and Gaussian curvature, for abstract Riemannian manifolds there are the sectional and the scalar curvature. Can one extend the curvature notions to polyhedra or to more general metric spaces?
In this talk I will discuss the Lipschitz-Killing curvatures of polyhedra and polyhedral manifolds, and the discrete Hilbert-Einstein functional in particular. Applications to the integral geometry and to rigidity will be presented.