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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).

• In $R^n$, there is a claim that it is enough to show that$$|\lambda A+(1-\lambda)B| \geq \lambda |A|+ (1-\lambda)|B|, \forall 0 \leq \lambda \leq 1$$ 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 see the MB inequality should look different:$$\phi(|A+B|) \geq \phi(|A|)+\phi(|B|)$$. Is there any idea of how $\phi$ should look like? question edit)

• 1

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).

1. In $R^n$, there is a claim that it is enough to show that $$|\lambda A+(1-\lambda)B| \geq \lambda |A|+ (1-\lambda)|B|, \forall 0 \leq \lambda \leq 1$$ 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?

2. 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.

3. 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?