Editing Distance geometry (section)

=== Embedding arbitrarily many points ===
The <math>n+3</math> case turns out to be sufficient in general.

In general, given a semimetric space <math>(R, d)</math>, it can be isometrically embedded in <math>\mathbb{R}^n</math> if and only if there exists <math>P_0, \ldots, P_n\in R</math>, such that, for all <math>k = 1, \ldots, n</math>, <math>(-1)^{k+1} \operatorname{CM}(P_0, \ldots, P_k) \ge 0</math>, and for any <math>P_{n+1}, P_{n+2} \in R</math>,

#<math>\operatorname{CM}(P_0, \ldots, P_n, P_{n+1}) = 0;</math>
#<math>\operatorname{CM}(P_0, \ldots, P_n, P_{n+2}) = 0;</math>
#<math>\operatorname{CM}(P_0, \ldots, P_n, P_{n+1},  P_{n+2}) = 0.</math>

And such embedding is unique up to isometry in <math>\mathbb{R}^n</math>.

Further, if <math>\operatorname{CM}(P_0, \ldots, P_n) \neq 0</math>, then it cannot be isometrically embedded in any <math>\mathbb{R}^m, m < n</math>. And such embedding is unique up to unique isometry in <math>\mathbb{R}^n</math>.

Thus, Cayley–Menger determinants give a concrete way to calculate whether a semimetric space can be embedded in <math>\mathbb{R}^n</math>, for some finite <math>n</math>, and if so, what is the minimal <math>n</math>.