# When Brunn-Minkowski inequatily $(m(A+B))^{1/d} \geq (m(A))^{1/d} + (m(B))^{1/d}$ becomes equality?

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

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

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

Edit:

As you might see, proving the inequality is easy, but find the necessary condition which makes the inequlity an equality is really hard. I searched for a long time about this, but most books (like Halmos's Measure Theory, Federer's Geometric Measure Theory etc) just give references instead of a proof and the proofs I found can date back to 1930s and are rather cmplicated.

Since E.M.Stein Real Analysis is a widely used book for analysis cource, I think if someone finished all problems in this book might have a better and modified proof. Thanks in advance.

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

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

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...