Defining a ball

Question

Which are appropriate phrases to define a ball?

1. Let $B$ be the open ball of radius $r$ centered at $x$.

2. Let $B$ be the open ball of radius $r$ around $x$.

3. Let $B$ be the open ball of radius $r$ about $x$.

4. Let $B$ be the open ball of radius $r$ with the center at $x$.

Answer 1

All of them are acceptable, in context, but you may want to refrain from $4$ and $2$. $4$ is just a wordy variation of $1$ and is unnecessary. The most common ways to phrase such a problem is through standard notation:

$$B_r (x)$$ is generally understood to be the open ball of radius $r$ centered at $x$.

Try to refrain from ambiguities or unnecessary wording.