The equality cases in the Brunn-Minkowski are:
- $A$ and $B$ lie in parallel hyperplanes (then all volumes are zero), or
- $A$ and $B$ are convex and homothetic.
A strengthening of the inequality should depend on some "non-homotheticity" measure of $A$ and $B$. There are some results in this direction, see Section 6.1 in Schneider's "Convex bodies: the Brunn-Minkowski theory", right after the proof of the BM inequality. See also Note 2 at the end of that section.