A Generalization of Costa’s Entropy Power Inequality

Aim of this short note is to study Shannon’s entropy power along entropic interpolations, thus generalizing Costa’s concavity theorem. We shall provide two proofs of independent interest: the former by
$\Gamma $
-calculus, hence applicable to more abstract frameworks; the latter with an explicit remainder term, reminiscent of Villani (IEEE Trans. Inf. Theory, 2006), allowing us to characterize the case of equality.