The following results are proved in Blumethal's book.

Embedding n + 1 points in the real numbers

Given a semimetric space , with , and , , an isometric embedding of into is defined by , such that for all .

Again, one asks whether such an isometric embedding exists for .

A necessary condition is easy to see: for all , let be the k-simplex formed by , then

The converse also holds. That is, if for all ,

then such an embedding exists.

Further, such embedding is