  The paper is centered around a new proof of the infinitesimal rigidity of
convex polyhedra. The proof is based on studying derivatives of the discrete
Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed
metric on the boundary. This approach is in a sense dual to using derivatives
of the volume in order to prove the Gauss infinitesimal rigidity of convex
polyhedra, which deals with deformations that preserve face normals and face
areas. In the spherical and in the hyperbolic-de Sitter space, there is a
perfect duality between the Hilbert-Einstein functional and the volume, as well
as between both kinds of rigidity. We also discuss directions for future
research, including elementary proofs of the infinitesimal rigidity of
hyperbolic (cone-)manifolds and development of a discrete Bochner technique.
