  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.
