Editing Distance geometry (section)

=== Embedding ''n'' + 1 points in the real numbers ===
Given a semimetric space <math>
 (S,d)</math> , with <math>S = \{P_0, \ldots, P_n\}</math>, and  <math>d(P_i, P_j) = d_{ij}\ge 0</math>, <math>0 \le i < j \le n</math>, an isometric embedding of <math>(S, d)</math> into <math>\mathbb{R}^n</math> is defined by <math display="inline">A_0, A_1,\ldots, A_n \in \mathbb R^n</math>, such that <math>d(A_i, A_j) = d_{ij}</math> for all <math>0 \le i < j \le n</math>.

Again, one asks whether such an isometric embedding exists for <math>(S,d)</math>.

A necessary condition is easy to see: for all <math>k = 1, \ldots, n</math>, let <math>v_k</math> be the ''k''-simplex formed by  <math display="inline">A_0, A_1,\ldots, A_k</math>, then

:<math>(-1)^{k+1} \operatorname{CM}(P_0, \ldots, P_k) = (-1)^{k+1} \operatorname{CM}(A_0, \ldots, A_k) = 2^k (k!)^k \operatorname{Vol}_k(v_k)^2 \ge 0</math>

The converse also holds. That is, if for all <math>k = 1, \ldots, n</math>,

:<math>(-1)^{k+1}\operatorname{CM}(P_0, \ldots, P_k) \ge 0,</math>

then such an embedding exists.

Further, such embedding is unique up to isometry in <math>\mathbb{R}^n</math>. That is, given any two isometric embeddings defined by <math>A_0, A_1,\ldots, A_n</math>, and <math>A'_0, A'_1,\ldots, A'_n</math>, there exists a (not necessarily unique) isometry <math>T :  \mathbb R^n \to \mathbb R^n</math>, such that <math>T(A_k) = A'_k</math> for all <math>k = 0, \ldots, n</math>. Such <math>T</math> is unique if and only if <math>\operatorname{CM}(P_0, \ldots, P_n) \neq 0</math>, that is, <math>A_0, A_1,\ldots, A_n</math> are affinely independent.