Distance geometry is the branch of mathematics concerned with characterizing and studying sets of points based only on given values of the distances between pairs of points. the n-dimensional volume of the simplex <math>v_n</math> satisfies
: <math> \operatorname{Vol}_n(v_n)^2 = \frac{(-1)^{n+1{(n!)^2 2^n} \operatorname{CM}(A_0, \ldots, A_n). </math>
Note that, for the case of <math>n=0</math>, we have <math>\operatorname{Vol}_0(v_0) = 1</math>, meaning the "0-dimensional volume" of a 0-simplex is 1, that is, there is 1 point in a 0-simplex.
<math display="inline">A_0, A_1,\ldots, A_n</math> are affinely independent iff <math>\operatorname{Vol}_n(v_n) > 0</math>, that is, <math> (-1)^{n+1} \operatorname{CM}(A_0, \ldots, A_n) > 0</math>. Thus Cayley–Menger determinants give a computational way to prove affine independence.
If <math>
k < n</math>, then the points must be affinely dependent, thus <math>
\operatorname{CM}(A_0, \ldots, A_n) = 0</math>. Cayley's 1841 paper studied the special case of <math>
k = 3, n = 4</math>, that is, any five points <math>
A_0, \ldots, A_4</math> in 3-dimensional space must have <math>
\operatorname{CM}(A_0, \ldots, A_4) = 0</math>.
History
The first result in distance geometry is Heron's formula, from 1st century AD, which gives the area of a triangle from the distances between its 3 vertices. Brahmagupta's formula, from 7th century AD, generalizes it to cyclic quadrilaterals. Tartaglia, from 16th century AD, generalized it to give the volume of tetrahedron from the distances between its 4 vertices.
The modern theory of distance geometry began with Arthur Cayley and Karl Menger. Cayley published the Cayley determinant in 1841, which is a special case of the general Cayley–Menger determinant. Menger proved in 1928 a characterization theorem of all semimetric spaces that are isometrically embeddable in the n-dimensional Euclidean space <math>\mathbb{R}^n</math>. In 1931, Menger used distance relations to give an axiomatic treatment of Euclidean geometry.
Leonard Blumenthal's book
Characterization via Cayley–Menger determinants
The following results are proved in Blumethal's book.