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

Abstract

We give a new proof, using only the Brascamp–Lieb inequality, of the fact that the Gaussian measure is the only strong log-concave measure having a strong log-concavity parameter equal to its covariance matrix. We unify the continuous and discrete settings by also giving a similar characterization of the Poisson measure in the discrete case, using “Chebyshev’s other inequality”. We briefly discuss how these results relate to Stein and Stein–Chen methods for Gaussian and Poisson approximation, and to the Bakry–Émery calculus.