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We study the connection between the concavity properties of a measure $\nu$and the convexity properties of the associated relative entropy $D(\cdot \Vert\nu)$ on Wasserstein space. As a corollary we prove a new dimensionalBrunn-Minkowski inequality for centered star-shaped bodies, when the measure$\nu$ is log-concave with a p-homogeneous potential (such as the Gaussianmeasure). Our method allows us to go beyond the usual convexity assumption onthe sets that appears essential for the standard differential-geometrictechnique in this area.  We then take a finer look at the convexity properties of the Gaussianrelative entropy, which yields new functional inequalities. First we obtaincurvature and dimensional reinforcements to Otto--Villani's ``HWI'' inequalityin the Gauss space, when restricted to even strongly log-concave measures. Ascorollaries, we obtain improved versions of Gross' logarithmic Sobolevinequality and Talgrand's transportation cost inequality in this setting.

New Brunn--Minkowski and functional inequalities via convexity of entropy