More refined versions of Brunn–Minkowski inequality and/or Prékopa–Leindler inequality

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.

• What about Brunn-Minkowski inequality but with essential minkowski sum? In this case $A+_{e}B=\{z \in \mathbb{R}^{n} : \mu(A \cap (\{z\}-B))\neq 0\}$. Clearly if $A$ or $B$ has $n$ dimensional Lebesgue measure zero then always $\mu(A \cap (\{z\}-B))=0$, and aessential minkowsi sum does not give you anything. So then we have just equality $| A+_{e}B|^{1/n}\geq |A|^{1/n}+|B|^{1/n}$ instead of inequality. The same thing with Prekopa--Leindler but with essential supremum. (see the last section en.wikipedia.org/wiki/Minkowski_addition) – Paata Ivanishvili Oct 9 '16 at 14:48

The equality cases in the Brunn-Minkowski are:

1. $A$ and $B$ lie in parallel hyperplanes (then all volumes are zero), or
2. $A$ 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.

It does not seem feasible to "refine" the format of the inequalities to "edge cases" you mentioned unless one wishes to lose the geometric flavor and elegance of the result. In fact, the B-M inequality fares better (as is done by some in the literature) when stated for (measurable) sets with non-empty interior.

Here is a good survey paper linking B-M to isometric inequalities:

http://www.ams.org/journals/bull/2002-39-03/S0273-0979-02-00941-2/S0273-0979-02-00941-2.pdf

• In my application, I essentially have a set-valued dynamical system, something like $A_{n+1} = f(A_n + B)$, and I wish to lower bound the measure of $A_n$ as $n \to \infty$. Brunn-Minkowski/Prekopa-Leindler gives lower bounds, but in some cases the bounds get weaker and weaker as $n$ gets large, so that it eventually is too weak to prove something I want. I'm hoping I can use additional information about the shapes of the sets, or something similar, to strengthen the lower bounds. – SorcererofDM Oct 9 '16 at 6:13