how is convexity defined in a subset $A \subset \{0,1\}^n$? furthermore, is there any extention of the Brunn-Minkowski inequality for subsets of $\{0,1\}^n$? thanks.

**Edit** (previously posted as an answer)
thank you for the reference article of Ollivier and Villani. I have a few misunderstanding though, regarding Brunn-Minkowski in $R^n$ and {0,1}$^n$(the hyperplane).

**Edit - this is the corrected question:**In $R^n$, there is a claim that it is enough to show that $$ |\lambda A+(1-\lambda)B| \geq |A|^{\lambda}|B|^{1-\lambda}, \forall 0 \leq \lambda \leq 1 $$ in order to conclude BM (Brunn-Minkowski) inequality: $$|A+B|^{1/n} \geq |A|^{1/n} + |B|^{1/n} $$ However, I couldn't think of how to prove it. Is it a trivial claim? How can someone prove it?In Ollivier and Villani's paper, it handles $M$, the middle points between a and b in the hypercube. I don't understand how can we expand this theory for $M'=\frac{1}{4}A+\frac{3}{4}B$, for example. we need it, I think, in order to conclude the real BM inequality in hypercube.

I feel there is a basic difference between BM inequality in $R^n$ and in the hypercube: in $R^n$ we claim $|A+B|^{1/n} \geq |A|^{1/n} + |B|^{1/n}$, with 1/n power-factor is quite intuitive since volume of balls in $R^n$ is $\sim r^n$. but balls in hyperplane don't grow that way... so, I assume the formula of the MB inequality should look different: $$ \phi(|A+B|) \geq \phi(|A|)+\phi(|B|) $$. Is there any idea of how $\phi$ should look like?