Sphere
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 that is considered to be the 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.

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

An octahedron is a sphere in taxicab geometry, and a cube is a sphere in geometry using the Chebyshev distance.
startend:quotation