Definition:Sphere

The $$n \ $$-dimensional sphere is the set

$$\mathbb{S}^n = \left\{{ x \in \R^{n+1} : |x-y|=r}\right\} \ $$

where $$r \in \R_+ \ $$ is called the radius of the sphere and $$y \in \R^{n+1} \ $$ is called the center of the sphere.

Frequently, the radius is taken as 1 and the center as the origin.