New Brunn--Minkowski and functional inequalities via convexity of entropy
This work addresses foundational problems in geometric and functional analysis, offering incremental advances by extending results beyond convexity assumptions and reinforcing existing inequalities in specific settings.
The paper tackles the problem of connecting concavity properties of measures with convexity of relative entropy in optimal transport, leading to a new dimensional Brunn--Minkowski inequality for star-shaped bodies under log-concave measures and improved functional inequalities like Gross' Logarithmic Sobolev and Talagrand's transportation cost inequalities for even strongly log-concave measures.
We study the connection between the concavity properties of a measure $ν$ and the convexity properties of the associated relative entropy $D(\cdot \Vert ν)$ along optimal transport. As a corollary we prove a new dimensional Brunn--Minkowski inequality for centered star-shaped bodies, when the measure $ν$ 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 Gauss space, when restricted to even strongly log-concave measures. As corollaries, we obtain improved versions of Gross' Logarithmic Sobolev inequality and Talagrand's transportation cost inequality in this setting.