I've been doing some research relating to the Brunn-Minkowski inequality, which states (among other things) that for $A,B$ convex bodies in $\mathbf{R}^d$ and $\mu$ the Lebesgue measure:
$$\mu(A+B)^{1/d} \geq \mu(A)^{1/d} + \mu(B)^{1/d}$$
with equality if and only if $A$ and $B$ are homothetic. I was curious if there are any good references for formulations of the inequality of the form:
$$\mu(A+B)^{1/d} = \mu(A)^{1/d} + \mu(B)^{1/d} + R(A,B)$$
where $R(A,B)$ is a "remainder term" which somehow records the failure of $A$ and $B$ to be homothetic. I've looked left and right, and unless I've missed something, I cannot find anything like this in the literature.