Equivalent Euclidean Data Complexes

Abstract

Euclidean data complexes are simplicial complexes that have been constructed from a point cloud in Euclidean space. Two of the most important examples of such complexes are the Čech and Alpha complex. In this thesis, we will prove that these are homotopy equivalent to the Delaunay-Čech complex using the geometric and gradient collapse arguments. Moreover, we introduce a new Euclidean data complex that we call the selective Delaunay-Alpha complex. Not only does it generalize the other three, but it is also simple-homotopy equivalent to them. The implications of this result will also be discussed.