  Collapsibility is a combinatorial strengthening of contractibility. We relate
this property to metric geometry by proving the collapsibility of any complex
that is CAT(0) with a metric for which all vertex stars are convex. This
strengthens and generalizes a result by Crowley. Further consequences of our
work are:
  (1) All CAT(0) cube complexes are collapsible.
  (2) Any triangulated manifold admits a CAT(0) metric if and only if it admits
collapsible triangulations.
  (3) All contractible d-manifolds ($d \ne 4$) admit collapsible CAT(0)
triangulations. This discretizes a classical result by Ancel--Guilbault.
