Are all Weakly Convex and Decomposable Polyhedral Surfaces Infinitesimally Rigid?

Abstract

It is conjectured that all decomposable (i.e. interior can be triangulated without adding new vertices) polyhedra with vertices in convex position are infinitesimally rigid and only recently has it been shown that this is indeed true under an additional assumption of codecomposability (i.e. the interior of the difference between the convex hull and the polyhedron itself can be triangulated without adding new vertices). One major set of tools for studying infinitesimal rigidity happens to be throughout the (negative) Hessian $M_T$ of the discrete Hilbert-Einstein functional. Besides its theoretical importance, it provides the necessary machinery to tackle the problem experimentally. To search for potential counterexamples to the conjecture, one constructs an explicit family of so-called $T$-polyhedra, all of which are weakly convex and decomposable, while being non-codecomposable. Since infinitesimal rigidity is equivalent to a non-degenerate $M_T$, one can let Mathematica search for the eigenvalues of $M_T$ and gather experimental evidence that such a flexible, weakly convex and decomposable $T$-polyhedron may not exist.