Are All Weakly Convex and Decomposable Polyhedral Surfaces Infinitesimally Rigid?

Authors

Jilly Kevo, University of Bonn, GermanyFollow

Abstract

It is conjectured that all decomposable (that is, 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 (that is, 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 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 \textit{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.

Author Bio

Jilly Kevo is an aspiring mathematician who resides in the Grand Duchy of Luxembourg. Besides her passion for struggling with a good math problem, she enjoys playing the piano and writing counterpoint. She also has regular exhibitions as a visual artist in her home country.

Faculty Sponsor

Jean-Marc Schlenker

Recommended Citation

Kevo, Jilly (2024) "Are All Weakly Convex and Decomposable Polyhedral Surfaces Infinitesimally Rigid?," Rose-Hulman Undergraduate Mathematics Journal: Vol. 25: Iss. 1, Article 6.
Available at: https://scholar.rose-hulman.edu/rhumj/vol25/iss1/6