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We prove, using the Brascamp-Lieb inequality, that the Gaussian measure isthe only strong log-concave measure having a strong log-concavity parameterequal to its covariance matrix. We also give a similar characterization of thePoisson measure in the discrete case, using "Chebyshev's other inequality". Webriefly discuss how these results relate to Stein and Stein-Chen methods forGaussian and Poisson approximation, and to the Bakry-Emery calculus.

An extremal property of the normal distribution, with a discrete analog