How to notate a set of point meeting a given condition?

Question

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

Answer 1

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 $\mathcal N_{n,r}(p)$.