  We prove that the second derived subdivision of any rectilinear triangulation
of any convex polytope is shellable. Also, we prove that the first derived
subdivision of every rectilinear triangulation of any convex 3-dimensional
polytope is shellable. This complements Mary Ellen Rudin's classical example of
a non-shellable rectilinear triangulation of the tetrahedron. Our main tool is
a new relative notion of shellability that characterizes the behavior of
shellable complexes under gluing.
  As a corollary, we obtain a new characterization of the PL property in terms
of shellability: A triangulation of a sphere or of a ball is PL if and only if
it becomes shellable after sufficiently many derived subdivisions. This
improves on results by Whitehead, Zeeman and Glaser, and answers a question by
Billera and Swartz.
  We also show that any contractible complex can be made collapsible by
repeatedly taking products with an interval. This strengthens results by
Dierker and Lickorish, and resolves a conjecture of Oliver. Finally, we give an
example that this behavior extends to non-evasiveness, thereby answering a
question of Welker.
