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The simplest case:

In the simplest case, shown in the first picture, we are given a finite set of points {p1... pn} in the Euclidean plane. In this case each site pk is simply a point, and its corresponding Voronoi cell Rk consists of every point in the Euclidean plane whose distance to pk is less than or equal to its distance to any other pk. Each such cell is obtained from the intersection of half-spaces, and hence it is a (convex) polyhedron.The line segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. The Voronoi vertices (nodes) are the points equidistant to three (or more) sites.

Properties:

• The dual graph for a Voronoi diagram (in the case of a Euclidean space with point sites) corresponds to the Delaunay triangulation for the same set of points.
• The closest pair of points corresponds to two adjacent cells in the Voronoi diagram.
• Assume the setting is the Euclidean plane and a group of different points is given. Then two points are adjacent on the convex hull if and only if their Voronoi cells share an infinitely long side.
• If the space is a normed space and the distance to each site is attained (e.g., when a site is a compact set or a closed ball), then each Voronoi cell can be represented as a union of line segments emanating from the sites. As shown there, this property does not necessarily hold when the distance is not attained.
• Under relatively general conditions (the space is a possibly infinite-dimensional uniformly convex space, there can be infinitely many sites of a general form, etc.) Voronoi cells enjoy a certain stability property: a small change in the shapes of the sites, e.g., a change caused by some translation or distortion, yields a small change in the shape of the Voronoi cells. This is the geometric stability of Voronoi diagrams.
• As shown there, this property does not hold in general, even if the space is two-dimensional (but non-uniformly convex, and, in particular, non-Euclidean) and the sites are points.

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