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 = 4) admit collapsible CAT(0) triangulations. This
discretizes a classical result by Ancel–Guilbault.