1. CJM 2013 (vol 66 pp. 783)
||Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert-Einstein Functional|
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.
The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed 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 review some of the related work and discuss directions for future research.
Keywords:convex polyhedron, rigidity, Hilbert-Einstein functional, Minkowski theorem