Geometric Complexes in Metric Spaces

A number of abstract simplicial complexes used in discrete and computational geometry, as well as in computational topology have gained prominence in the recent years. These complexes are defined starting from some data in a metric space. Strictly speaking, in current applications the input data is not combinatorial, but rather geometric. However, due to a profusion of metric spaces in combinatorial contexts, we include the definitions of these families of abstract simplicial complexes. 

Definition 9.12. Let M he a metric space, and assume that we are given a set S of points in M and a nonnegative real number t. The Rips complex Rt S) is the abstract simplicial complex whose set of vertices is S, and a C S is a simplex if and only if the distance between any two points in a does not exceed t. 

Sometimes the complex Rt S) is called Vietoris-Rips complex. A possible variation on the Rips complex is to take as simplices those sets of vertices that are contained in a closed ball with radius t. 

Consider now a collection of closed balls B = in a metric space 

where Bi = ci,r ), the Cj s are their centers, and the rfs are their radii. For each ball B we define its weighted Voronoi region as 

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