On a Brunn-Minkowski principle for first-order information functionals
In this work we present the following general principle: every positive homogeneous functional on an abstract convex cone satisfying a suitable Brunn-Minkowski inequality gives rise to an information functional -defined as the quotient of the functional by its relative surface area- that satisfies a linear Brunn-Minkowski inequality. We also establish the dual counterpart of this principle for functionals satisfying an analogous convexity property, yielding reverse linear Brunn-Minkowski inequalities for the associated information functionals. Furthermore, we show that homogeneity is necessary within this principle by providing a counterexample to the expected result for the information functional associated to the standard Gaussian measure. As applications, we recover, via a different, first-order argument, a classical result by Giannopoulos, Hartzoulaki and Paouris (2002) for the ratio of two consecutive quermassintegrals, together with a characterization of the equality case. The same argument extends to the setting of mixed discriminants of symmetric positive definite matrices. Finally, our approach also yields a new $L_p$ extension ($p \geq 1$) of the Brunn-Minkowski inequality by Dembo, Cover and Thomas for the classical information functional, where the relative surface area is expressed in terms of Lutwak's first $L_p$ quermassintegral.