A concavity property for the reciprocal of Fisher information and its consequences on Costa's EPI

We prove that the reciprocal of Fisher information of a log-concave probability density X in Rn is concave in t with respect to the addition of a Gaussian noise Zt=N(0,tIn). As a byproduct of this result we show that the third derivative of the entropy power of a log-concave probability density X in Rn is nonnegative in t with respect to the addition of a Gaussian noise Zt. For log-concave densities this improves the well-known Costa’s concavity property of the entropy power (Costa, 1985).

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