Isometric embeddings of snowflakes into finite-dimensional Banach spaces

We consider a general notion of snowflake of a metric space by composing the distance with a nontrivial concave function. We prove that a snowflake of a metric space XX isometrically embeds into some finite-dimensional normed space if and only if XX is finite. In the case of power functions we give a uniform bound on the cardinality of XX depending only on the power exponent and the dimension of the vector space.