We study the connection between the concavity properties of a measure $\nu$ and the convexity properties of the associated relative entropy $D(\cdot \Vert \nu)$ along optimal transport. 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 is fundamentally 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ν的凹性性质与相关相对熵D(·∥ν)沿最优传输的凸性性质之间的联系。作为推论，当测度ν具有p-齐次势（如高斯测度）且为对数凹时，我们证明了中心星形体的一个新维度Brunn-Minkowski不等式。我们的方法突破了该领域标准微分几何技术对集合凸性这一基础性前提假设的限制。随后，我们对高斯相对熵的凸性进行了更精细的分析，得到了新的泛函不等式。首先，当限制于偶强对数凹测度时，我们在高斯空间中得到了Otto-Villani“HWI”不等式的曲率与维度强化版本。作为推论，在此设定下我们获得了Gross对数Sobolev不等式和Talagrand运输代价不等式的改进形式。