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?

An observation is that 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 guidance or feedback would be deeply appreciated. Thank you very much.

Suppose we have two points $p_1$ and $p_2$ in a metric space with 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?

An observation is that 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 guidance or feedback would be deeply appreciated. Thank you very much.