When does Brunn-Minkowski inequatily $(m(A+B))^{1/d} \geqslant (m(A))^{1/d} + (m(B))^{1/d}$ become equality?

Question

Let $A$ and $B$ be two non-empty compact subsets of $\mathbb{R}^d$, $m$ means Lebesgue measure.

Brunn-Minkowski inequality gives $(m(A+B))^{1/d} \geqslant (m(A))^{1/d} + (m(B))^{1/d}$.

But how to prove the following?

$(m(A+B))^{1/d} = (m(A))^{1/d} + (m(B))^{1/d} \implies$

$A$ and $B$ are convex and $\exists$ $\delta＞0$, $h\in \mathbb{R}^d$ s.t. $A = \delta B + h$

(It's Problem $8$ of Chapter $1$, Page $48$ in E.M.Stein's Real Analysis , though this problem doen't have much to do with real analysis.)

Thanks in advance.

Answer 1

You might also be interested in the following two papers by Fields medalist A. Figalli:

• Quantitative stability for the Brunn-Minkowski inequality,
• Stability result for the Brunn-Minkowski inequality.

The stability result, proven with optimal transport techniques, implies in particular the equality condition.

Answer 2

Some proofs I found, which are really complicated since they contain a lot of definitions and lemmas new to others. Moreover, it's impossible to answer this question only in a few words, so I just list some references here for those who really need.

$(1)$ Rolf Schneider, Convex Bodies: the Brunn-Minkowski Theory. Section $6.1$, the Brunn-Minkowski theorem.

$\quad$Since this book mainly concentrates on convex bodies, it only proves for convex bodies when the inequality becomes equality without showing why it's not ture for non-convex sets.

$(2)$ IMRE Z. RUZSA, The Brunn–Minkowski Inequality and Nonconvex Sets.

$\quad$This article improves the inequlity for convex sets in $\mathbb R^n$.

$(3)$ DANIEL A. KLAIN, ON THE EQUALITY CONDITIONS OF THE BRUNN-MINKOWSKI THEOREM.

$\quad$This article describes a new proof of the equality condition for convex sets in $\mathbb R^n$.

To be continued...