Sphere - Generalization To Metric Spaces

Generalization To Metric Spaces

More generally, in a metric space (E,d), the sphere of center x and radius r > 0 is the set of points y such that d(x,y) = r.

If the center is a distinguished point considered as origin of E, as in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the case of a unit sphere.

In contrast to a ball, a sphere may be an empty set, even for a large radius. For example, in Zn with Euclidean metric, a sphere of radius r is nonempty only if r2 can be written as sum of n squares of integers.