When Convexity Helps Collapsing Complexes

This paper illustrates how convexity hypotheses help collapsing simplicial complexes. We first consider a collection of compact convex sets and show that the nerve of the collection is collapsible whenever the union of sets in the collection is convex. We apply this result to prove that the Delaunay complex of a finite point set is collapsible. We then consider a convex domain defined as the convex hull of a finite point set. We show that if the point set samples sufficiently densely the domain, then both the Cech complex and the Rips complex of the point set are collapsible for a well-chosen scale parameter. A key ingredient in our proofs consists in building a filtration by sweeping space with a growing sphere whose center has been fixed and studying events occurring through the filtration. Since the filtration mimics the sublevel sets of a Morse function with a single critical point, we anticipate this work to lay the foundations for a non-smooth, discrete Morse Theory.