Brunn-Minkowski inequality lower bounds the measure of a Minkowski sum by the measures of the summands. Its statement reads as follows:

Let $n$ ≥ 1 and let $μ$ denote the Lebesgue measure on $\mathbb R^n$. Let $A$ and $B$ be two nonempty compact subsets of $R^n$. Then

$$ \mu (A + B)^{1/n} \geq \mu (A)^{1/n} + \mu (B)^{1/n},$$

where $A + B$ denotes the Minkowski sum:

$$A + B := \{\, a + b \in \mathbb{R}^{n} \mid a \in A,\ b \in B\,\}.$$

However, the inequality loses a lot of strength if one or both of $A$ and $B$ is thin along some direction. For example, in $\mathbb R^2$, if $A$ is the segment connecting $(0, 0)$ to $(0, 1)$, and $B$ is the segment connecting $(0, 0)$ to $(1, 0)$, then $A + B$ is the square with corners $(0, 0), (0, 1), (1, 1), (1, 0).$ In the Brunn-Minkowski inequality, the LHS would be 1 but the RHS would be 0, and the "slack" in the inequality is very large.

I'm wondering if we know of any "refinement" of Brunn-Minkowski inequality or its integral form the Prekopa-Leindler inequality that would fare better in edge cases like the above, where some of the summands are "thin." Both inequalities are tight as they are usually stated, so I'm expecting we would need additional data to compute correction factors, perhaps something related to the shapes of the sets.