Up front two remarks to avoid any confusion and misconceptions. First, it is more cumbersome to distinguish between d=2 and d=3 dimensions than to phrase all definitions in , for some arbitrary but fixed positive integer d. Second, we use square roots of real numbers as weights. Since negative weight squares do make sense, we choose as the domain for all point weights. This is the set of all (positive) square roots of real numbers, and it inherits its linear order from .
A point with weight , is interpreted as a spherical ball,
with center p and radius , where |yz| denotes the Euclidean distance between points y and z. All points with non-real weight correspond to empty balls.
Figure 1: The union of a set of disks in the plane.
The shape of a finite set of weighted points is defined in terms of a decomposition of the union of balls, , into convex sets (see, for example, figures 1 and 2).
Figure 2: The decomposition of the union of disks defind by their Voronoi cells.
The (weighted) Voronoi cell of a ball is
It is a convex polyhedron, and its intersection with the ball union is convex because . Note that the convex cells have pairwise disjoint interiors, but some of them overlap along common boundary pieces. These pieces of overlap are instrumental in the construction of a set system closed under the subset operation. In topology, such a system is referred to as an abstract simplicial complex. Specifically, we define the nerve of as
Assuming general position of the points or balls, the largest set in is of size d+1, i.e., a triple in and a quadruple in . Under this assumption, has a natural geometric realization by mapping each cell to the point . This realization is a (geometric) simplicial complex, , see e.g. . Each set is represented by the convex hull of the corresponding points: the points are the images of the cells in X and their convex hull is a simplex of dimension one less than the cardinality of X. Formally,
Figure 3: The dual complex of the disk union.
We refer to this complex as the dual complex of , and to its underlying space, , as the dual shape (see figures 3 and 4).
Figure: The decomposition of the union of disks and its dual complex (figures 2 and 3 overlapped).
Among the most useful properties of are the homotopy equivalence between and , and the fact can be expressed as the alternating sum of common ball intersections, with one term per simplex in . This implies short inclusion-exclusion formulas for the d-dimensional volume and other measures of , see .