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It is proved that a continuous function f on ℝ n is negative definite if and only if it is polynomially bounded and satisfies the inequality E f ( X - Y ) ≤ E f ( X + Y ) for all i.i.d. random vectors X and Y in ℝ n . The proof uses Fourier transforms of tempered distributions. The "only if" part has been proved earlier by Lifshits et al. (A probabilistic inequality related to negative definite functions.