Barycentric subdivisions of convex complexes are collapsible

A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d<4. We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in ${\mathbb{R}}^d$ is simplicially collapsible after d-2 derived subdivisions at most. This presents progress on an old question by Goodrick.