A centroaffine perspective on improved Brascamp--Lieb and dimensional Brunn--Minkowski inequalities

We study dimensional concavity properties of even log-concave measures within the framework of weighted centroaffine differential geometry. We express the relevant local concavity inequalities in terms of a weighted Hilbert--Brunn--Minkowski operator and establish an anisotropic Poincaré inequality for odd functions and a strengthened Brascamp--Lieb inequality for even functions. As applications, we obtain quantitative dimensional concavity estimates for a class of geometric measures and measures with pinched Hessian, extending results of Cordero-Erausquin and Rotem for rotationally invariant measures and Eskenazis and Moschidis for the Gaussian measure.