**Goal:**

A classical results of Legendre and Cauchy states that convex polyhedra are rigid: they are uniquely determined by the "shape" of their faces. They are also *infinitesimally rigid*, that is, any non-trivial first-order deformation induces a non-zero first-order deformation of the shape of a face.

More recently, it was conjectured that this infinitesimal rigidity property extends to *weakly convex* polyhedra, that is, polyhedra which are not convex but have the same vertex set as a convex polyhedron, under an additional condition of *decomposability*: a polyhedron is decomposable if it can be cut into convex pieces without introducing any additional vertex.

This infinitesimal rigidity conjecture turns out to be true (see [1]) under an additional condition of *codecomposability* (basically, the complement of a polyhedron in its convex hull is decomposable). The main goal of the project would be to explore whether this additional condition is really necessary.

An intermediate step, or tool, would be to compute a certain symmetric matrix associated to decomposable polyhedron, and explore the index of this matrix.

**Supervisor:**
Jean-Marc Schlenker

**Difficulty level:**
Medium/Advanced.

**Tools:**

- SageMath

**References:**

- [1] Izmestiev, Ivan and Schlenker, Jean-Marc, Infinitesimal rigidity of polyhedra with vertices in convex position. Pacific J. Math. 248 (2010), no. 1, 171–190.