Stoker's Conjecture

Stoker's conjecture, stating that all dihedral angles together determine the shape of a simplicial convex polytope, is one of various rigidity problems in polytope geometry. We consider local rigidity, meaning that within permited small changes, whether dihedral angles can determine the shape. Building a group of equations, we know that: all convex polytopes with prescribed cominatorial structure form a manifold; in sight of manifold theory, it suffices to prove the rank of the Jacobian matrix associated to that group of equations equals to a special number. Under the further assumption that all faces are acute-angle-triangles, it can be converted to a graphical problem. Utimately, we prove that convex polytopes with all faces acute-angle-triangles and with no more than 10 faces satisfy local rigidity.From local rigidity to manifold, to linear algebra, and then to graph theory, the abundance and complexity lying in convexity is reflected; it's also the way that we simplify the problem.