# Defining a ball in mathematics

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 $x$.

Or, without \$,

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.

Remark I mean a ball in an arbitrary metric space.

• In a math text you wouldn't say "ball", you'd say "sphere". I'm not sure what you mean by an "open" ball.
– Jay
Aug 29, 2014 at 13:45
• @snailplane MathJax is a site-wide setting, and the community needs to justify it with examples of how it would be useful. I doubt it would be enabled for ELL. Aug 30, 2014 at 5:05
• @Jay Open ball is a technical term for the space within a sphere but excluding the surface of the sphere. Aug 30, 2014 at 5:08
• I'm voting to close this question as off-topic because this is a question of technical jargon, not general English. Nov 16, 2019 at 21:43

Since a ball is a three dimensional object, to define it you need to specify its radius and its center.

Its center is a vector (also three dimensions). You can show this using typography or words:

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

Most authors would define a standard typography to indicate scalars and vectors and use that throughout their textbook.

• I don't want to get into a different question, but I wouldn't say that a ball or sphere is centered at a "vector", but rather at a "point". "Let B be the sphere of radius r centered at the point p", or for a specific case, "Let B be the sphere of radius 3 centered at the point (4, 7, -2)".
– Jay
Aug 29, 2014 at 13:52
• @Jay I agree, especially if it is clear from the context that the point is in 3-D space. Aug 29, 2014 at 14:09
• Actually, I meant to ask about a ball in an arbitrary metric space. Aug 29, 2014 at 15:59
• @Sinusx Then see Jay's comments, you may want to add a phrase about the multi-dimensional nature of the center point if it is not obvious from the context. Aug 29, 2014 at 16:05