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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.

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    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 '14 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. – 200_success Aug 30 '14 at 5:05
  • @Jay Open ball is a technical term for the space within a sphere but excluding the surface of the sphere. – 200_success Aug 30 '14 at 5:08
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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.

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    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 '14 at 13:52
  • @Jay I agree, especially if it is clear from the context that the point is in 3-D space. – Gary's Student Aug 29 '14 at 14:09
  • Actually, I meant to ask about a ball in an arbitrary metric space. – Sinusx Aug 29 '14 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. – Gary's Student Aug 29 '14 at 16:05

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