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.