A note on positive definite norm dependent functions

Abstract

Let K be an origin symmetric star body in ℝⁿ. We prove, under very mild conditions on the function f : [0, ∞)→ℝ, that if the function f(‖x‖_K) is positive definite on ℝⁿ, then the space (ℝⁿ, ‖ ⋅ ‖_K) embeds isometrically in L₀. This generalizes the solution to Schoenberg’s problem and leads to progress in characterization of n-dimensional versions, i.e. random vectors X=(X₁, …, X_n) in ℝⁿ such that the random variables ∑a_iX_i are identically distributed for all a∈ℝⁿ, up to a constant depending on ‖a‖_K only.