Cayley-Menger determinants

These polynomials tend to be rather complicated, and hence it is fortunate that they can be written simply in terms of Cayley-Menger determinants. To see how this works, consider the inner square of a single pair of interpoint vectors x = b — a... 

This last determinant can be rewritten as a three-point Cayley-Menger determinant ... 

In the following, we shall denote such a symmetric three-point Cayley-Menger determinant by D(a, b, c). The general definition of an W-point Cayley-Menger determinant is... 

Thus a four-point Cayley-Menger determinant is equal to the product of the signed volumes spanned by the parallelepipeds whose sides are the vectors from one of the points to the other three (in each of the two point sets separately). This is the syzygy that connects the squared distances to the signed volumes. The five-point Cayley-Menger determinants, on the other hand, are the Gramians of four interpoint vectors, and hence vanish identically whenever the distances are three-dimensional, as described at the beginning of this section. 

The tetrangle inequality is a consequence of the nonnegativity of the symmetric four-point Cayley-Menger determinants. These do not factorize like the three-point determinants, but they can be expanded as a quadratic function of one of the squared distances, e.g., D(3,4), as follows ... 

---