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 dimensional Brunn-Minkowski inequality for centered star-shaped bodies, when the measure $\nu$ is log-concave with a p-homogeneous potential (such as the Gaussian measure). Our method allows us to go beyond the usual convexity assumption on the sets that appears essential for the standard differential-geometric technique in this area. We then take a finer look at the convexity properties of the Gaussian relative entropy, which yields new functional inequalities. First we obtain curvature and dimensional reinforcements to Otto--Villani's ``HWI'' inequality in the Gauss space, when restricted to even strongly log-concave measures. As corollaries, we obtain improved versions of Gross' logarithmic Sobolev inequality and Talgrand's transportation cost inequality in this setting.

翻译：我们研究了测度$\nu$的凹性性质与Wasserstein空间上关联的相对熵$D(\cdot \Vert \nu)$的凸性性质之间的联系。作为推论，当测度$\nu$具有$p$次齐次势（如高斯测度）且为对数凹测度时，我们证明了中心星形体上新的维数Brunn–Minkowski不等式。我们的方法能够突破该领域中标准微分几何技术所依赖的集合通常凸性假设。随后，我们对高斯相对熵的凸性性质进行了更精细的分析，得出新的泛函不等式。首先，当限制在偶强对数凹测度时，我们得到了高斯空间中Otto–Villani "HWI"不等式的曲率与维数强化形式。作为推论，我们在该设定下获得了Gross对数Sobolev不等式和Talagrand运输成本不等式的改进版本。