Suppose we have two points $p_1$ and $p_2$ in a metric space with $\bf{unknown}$ dimensionality, with no way to directly compute the distance between them, e.g. no coordinates.
Say we can randomly sample a set of points $P$ in this space. And say we can calculate the distance between any pair of points in $P$, the distance between any point in $P$ and $p_1$, and the distance between any point in $P$ and $p_2$.
The question I have is: can the distance between $p_1$ and $p_2$ be estimated using points in $P$ and the distances we can calculate with them? Is there some triangulation scheme that would work, without knowing dimensionality? Even if the distance cannot be calculated with certainty is there some statistical estimate? Relatedly, are there restrictions we can place on the space that make this problem tractable?
Any guidance or feedback would be deeply appreciated. Thank you very much.
Update 1: The best I can think is the difference in the distances between a point in $P$ and points $p_1$ and $p_2$ is a lower bound on the distance between $p_1$ and $p_2$. But this seems like a very weak bound. Any further thoughts would be appreciated. Any direct estimate would be amazing.
Update 2: A concrete example is as follows. Say there are two satellites and we can measure distances between known ground locations and each satellite. What is the distance between the satellites? The one difference is we want a procedure that works regardless of the dimensionality of the space.