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

Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional I: Polyhedral case