Collapsibility is a classical notion introduced by Whitehead as part of his simple homotopy theory. We provide several results relating it to metric geometry and convexity.
(1) Every complex that is CAT(0) with a metric for which all vertex stars are convex is collapsible.
Focus Area 3: Topological connectivity and diameter of Discrete Structures
(2) Any linear subdivision of any polytope is simplicially collapsible after one barycentric subdivision. This solves up to one derived subdivision a classical question by Lickorish.
(3) Any linear subdivision of any star-shaped polyhedron in R^s is simplicially collapsible after d-2 barycentric subdivisions at most. This presents progress on an old question by Goodrick.
We furthermore provide the following applications:
(1) Any simplicial complex admits a CAT(0) metric if and only if it admits collapsible triangulations.
(2) All contractible manifolds (except for some 4-dimensional ones) admit collapsible CAT(0) triangulations. This provides a polyhedral version of a classical result of Ancel and Guilbault.
(3) There are exponentially many geometric triangulations of S^d. This interpolates between the known result that boundaries of simplicial (d+1)-polytopes are exponentially many, and the conjecture that d-spheres are more than exponentially many.
(4) In terms of the number of facets, there are only exponentially many geometric triangulations of space forms with bounded geometry. This establishes a discrete version of Cheeger's finiteness theorem.