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How do you formulate the set of the n closest neighbours within the radius r of point P with proper mathematical syntax?

Simen Musæus
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  • And what if there are several points the same distance away from $P$, so that none of them are "closest"? – complexist May 25 '14 at 19:48
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    Assuming $X$ being is set of all points, you could go for something like this: $$ S = \arg\max {\lvert S^\prime\rvert \mid S^\prime\subseteq X\cap \mathcal{B}(P,r), \lvert S^\prime\rvert \leq n} $$ If the set you are trying to get is well-defined, this would work; if there are no such set (too many points at equal distance from $P$, or not enough in the open ball of radius $r$), then $S$ will be either empty or will contain less than $n$ points. – Clement C. May 25 '14 at 19:52
  • @ClementC. Is $\arg\max $ understood as empty set when the maximum is not unique? I though it would be the set of maxima, i.e., sets of sets in this case. Wikipedia also thinks so. –  May 25 '14 at 19:56
  • Yes, that would be an issue indeed if there are several sets achieving this maximum. I don't know how to formalize it, though, without the assumption (implicit in the OP's question) that there is only one such set... – Clement C. May 25 '14 at 20:03

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There is no notation that will do the job for you; you have to explain the concept to the reader first. When doing so, you'll have to address corner cases:

  • some points have equal distance to $P$, so there is no unique set of "closest $n$"
  • The number of points within radius $r$ of $P$ is less than $n$.

Once you decided how to treat these cases, and communicated your decision to the reader in words, you can give the set any notation you want, such as $\mathcal N_{n,r}(p)$.