I am given a finite set of geometric points in three dimensions. I want to add another point $A$, so that it's close to a certain point $P$ (that is $d(P, A)\lt k$, where k is an arbitrary constant), but distant from the others ($d(P, X)\gt k$, where $X$ is any other point).
I am clueless. Numerical solutions or fallible algorithms are welcome.
One simplification I thought is that one could look for a certain surface passing through $P$, so that the other points are all on one side of the subdivided space, then place $A$ to the other side. This can cut off some solutions to the first problem, but since in my actual situation the point positions have some uncertainty, a stricter solution leaves the margin of error that I need. Unfortunately, I didn't make up a good method to calculate this surface (rather a set of) which solve the problem.