Suppose $A,B \subset \mathbb{R}^d$ are convex sets such that $m(A+B)^{1/d} = m^{1/d}(A)+m^{1/d}(B)$. Show that there exists $\lambda>0$ and $x \in \mathbb{R}^d$ such that $A=\lambda B + x$.
Based on Wiki (https://en.wikipedia.org/wiki/Brunn%E2%80%93Minkowski_theorem), I know this is true but don't know how to prove it.
My effort: I have proved the case when $A$ and $B$ are both $d$-dimensional rectangles. Next, I think I can move on to a step that $A$ and $B$ are union of finitely many rectangles. If I do that, it means I can say this is true for open sets $A$ and $B$ of finite measure. Can you evaluate my effort? or suggest a different path which is possibly easier?