Definition:Sphere/Normed Vector Space/Radius
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Definition
Let $\struct{X, \norm{\,\cdot\,}}$ be a normed vector space.
Let $x \in X$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {S_\epsilon} x$ be the $\epsilon$-sphere of $x$.
In $\map {S_\epsilon} x$, the value $\epsilon$ is referred to as the radius of the $\epsilon$-sphere.
Linguistic Note
The plural of radius is radii, pronounced ray-dee-eye.
This irregular plural form stems from the Latin origin of the word radius, meaning ray.
The ugly incorrect form radiuses can apparently be found, but rarely in a mathematical context.