Some remarks on Euclidean invariants

Summary.

Let E denote the n-dimensional Euclidean space,$n \geq 2$. By a Euclidean invariant we mean a real-valued function f defined on a subset of $E^{m+1}$ such that for all Euclidean similarity transformations g the functional equation $f(P_0,P_1,\dots ,P_m) = f(g(P_0),g(P_1),\dots ,g(P_m))$ is satisfied. It is to be understood here that the domain of definition of f is invariant under all g. For instance, functions which are expressible as quotients of homogeneous polynomials of equal degree in the distances $u_{ij} = d(P_i,P_j)$ satisfy these requirements. We call them rational invariants. It is shown conversely, that if the bijective function $g: E \to E$ is in both directions differentiable in class $C^1$ and if f is a non-trivial rational invariant such that $m \leq n$, the domain of f is invariant under g, and $f(P_0,P_1,\dots ,P_m) = f(g(P_0),g(P_1),\dots ,g(P_m))$ holds identically on the domain of f, then g is a Euclidean similarity transformation. An analogous statement is true, if f is a continuous Euclidean invariant which is not constant on m-simplexes.