Infinitesimal rigidity of convex polyhedra through the second derivative of the Hilbert-Einstein functional
Let \(P\) be a compact convex polyhedron in \(\mathbb R^3\) whose vertices are denoted \(p_i\). For small \(t\), let \(P(t)\) be a deformation of \(P\). This is a polyhedron with the same combinatorics as \(P\) and vertices \(p_i+tq_i\) for a given set of vectors \(q_i\). The deformation is called isometric if the edge lengths of \(P(t)\) remain constant in the first order at \(t=0\).NEWLINENEWLINEThe author provides a new proof of Dehn's theorem stating that every convex polyhedron \(P\) in \(\mathbb R^3\) is infinitesimally rigid in the sense that every isometric deformation of \(P\) is trivial. The proof employs derivatives of the discrete Hilbert-Einstein functional on the space of warped polyhedra with a fixed boundary metric.
scientific article; zbMATH DE number 6331498