Triangulating Topological Spaces

Given a subspace [Formula: see text] and a finite set S⊆ℝd, we introduce the Delaunay complex, [Formula: see text], restricted by [Formula: see text]. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets [Formula: see text] in a non-empty set. By the nerve theorem, [Formula: see text] and [Formula: see text] are homotopy equivalent if all such sets are contractible. This paper proves a sufficient condition for [Formula: see text] and [Formula: see text] be homeomorphic.

PERIODIC AND QUASI-PERIODIC LAGUERRE TILINGS

The Laguerre construction extends and unifies the notions of Voronoi (or Dirichlet) and Delaunay complex. It is shown how the standard projection formalism for the generation of quasi-periodic tilings from higher-dimensional periodic “oblique” (or “klotz”) tilings associated to periodic Voronoi and Delaunay complexes also applies more generally to Laguerre complexes. It turns out that all quasi-periodic tilings obtained this way are Laguerre tilings themselves.