Multidimensional and non-microstates extensions of the free Brunn–Minkowski inequality

Develop a multidimensional analogue of the one-dimensional free Brunn–Minkowski (Prekopa–Leindler) inequality of Ledoux–Popescu and, additionally, formulate and prove a corresponding extension in the non-microstates framework of free entropy/pressure, where random matrix approximation is unavailable.

Background

The paper recalls that the free Brunn–Minkowski inequality is currently known only in one dimension, via random matrix approximation (Ledoux) and mass transport (Ledoux–Popescu). Extending this inequality beyond one dimension would parallel classical convex geometry where multidimensional versions underpin functional inequalities.

The authors emphasize that a non-microstates extension would be even more challenging, since matrix approximation techniques no longer apply in that setting. Such an extension would be significant for proving free functional inequalities without relying on microstates methods.