Why the Bonnesen inequality is better than the Brunn-Minkowski one?

Question

I'm trying to understand why the Bonnesen inequality is better than the Brunn-Minkowski one. The first one says that, if $A$ and $B$ are convex bodies in $\mathbf{R}^n$ and $H$ is a $n-1$-dimensional subspace, then $$\mu(A+B)\geq \left(a^{\frac{1}{n-1}}+b^{\frac{1}{n-1}}\right)^{n-1}\left(\frac{\mu(A)}{a}+\frac{\mu(B)}{b}\right),$$ where $a=\mu(\phi_H(A))$ and $b=\mu(\phi_H(B))$, while the Brunn-Minkowski formula says that: $$\mu(A+B)\geq(\mu(A)^{1/n}+\mu(B)^{1/n})^n$$

Answer 1

By Holder's inequality with $p=n$ and $q=\frac n {n-1}$ you can see that RHS of Brunn-Minkowski inequality is less than or equal to RHS of Bonnesen inequlity.

Let $x=\frac {\mu(A)} a,y=\frac {\mu(B)} b$. Then $\mu(A)^{1/n}+\mu(B) ^{1/n} =a^{1/n}x^{1/n}+b^{1/n}y ^{1/n}\leq (a^{q/n}+(b^{q/n})^{1/q} (x+y)^{1/n}$. Hence $(\mu(A)^{1/n}+\mu(B) ^{1/n} )^{1/n} \leq (x+y) (a^{n-1}+(b^{n-1})^{1/q}$. Thus Bennesen inequality implies Brunn _ Minkowski inequality.