Dimension of a snowflake of a finite Euclidean subspace

Let $X$ be an $n$-point subset of a Euclidean space and $0 < a < 1$. The classical theorem of Schoenberg implies that the snowflake space $X^a$ can be isometrically embedded into Euclidean space. In the paper we show that points in the image of such an embedding always are in general position. As application we prove the analogue of Schoenberg's result for quotients of Euclidean spaces by finite groups.