Definition:Sphere

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

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

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

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