The Equivalence of Brunn-Minkowski Inequalities

Question

I am trying to make a proof on Brunn-Minkowski inequalities. Specifically, I am trying to prove that given two convex bodies C and D in $\mathbb{R}^n$, then the Brunn-Minkowski inequality

$Vol[(\lambda C + (1-\lambda)D)]^\frac{1}{n} \geq \lambda[Vol(C)]^\frac{1}{n} + (1-\lambda)[Vol(D)]^\frac{1}{n}$ for all $\lambda \in [0,1]$

holds if and only if

$Vol[(C + D)]^\frac{1}{n} \geq [Vol(C)]^\frac{1}{n} + [Vol(D)]^\frac{1}{n}$

I had thought of this from the context of Minkowski sum but I haven't been able to convince myself in that regard. I can as well prove the other forms of BM inequalities but this particular proof eludes me

I would appreciate any help as regards this proof.

Thanks in anticipation.

Answer 1

For the forward direction, take $\lambda = \frac{1}{2}$. For the backward direction, observe that

$$\lambda \mathop{Vol}(C)^{1/n} = \mathop{Vol}(\lambda C)^{1/n}$$