Triangulating Topological Spaces

Herbert Edelsbrunner, Nimish R. Shah

Abstract: Given a subspace 𝒳 ⊆ Rd and a finite set S⊆Rd, we introduce the Delaunay simplicial complex, D𝒳, restricted by 𝒳. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets 𝒳 in a non-empty set. By the nerve theorem,⋃D𝒳 and 𝒳 are homotopy equivalent if all such sets are contractible. This paper shows that ⋃D𝒳 and 𝒳 are homeomorphic if the sets can be further subdivided in a certain way so they form a regular CW complex.