On Pg. $33$ of these notes (a), the author says (somewhere in the middle of the page) that it suffices to prove $$\operatorname{vol} (\partial\tilde K) \le 2n \operatorname{vol}(\tilde K)^{(n-1)/n}$$ in order to prove the symmetric version of Theorem $6.1$ using Theorem $6.2$. Why is this true?
The author also remarks that $\operatorname{vol}(\tilde K) \le 2^n$, if $\tilde K$ is such that its maximal volume ellipsoid is $B^n_2$. Why is this true? This is also used later in the proof of Theorem $6.2$.
Reference (b) has some details but I could not find anything that answers my question.
For reference:
Symmetric case of Theorem $6.1$:
Let $K$ be a convex body and $Q$ a cube in $\mathbb R^n$. Then, there is an affine image of $K$ whose volume is the same as that of $Q$ and whose surface area is no larger than that of $Q$.
Theorem $6.2$:
Among symmetric convex bodies the cube has the largest volume ratio.
where the volume ratio $\operatorname{vr}(K)$ for convex bodies is defined as $$\operatorname{vr}(K) = \frac{\operatorname{vol}(K)}{\operatorname{vol}(\mathcal E)}$$ where $\mathcal E$ is the maximal volume ellipsoid contained in $K\subset\mathbb R^n$.
My thoughts: I understand that if $\operatorname{vol}(\tilde K) = \operatorname{vol}(Q)$, then $\operatorname{vol}(\partial\tilde K) \le \operatorname{vol}(\partial Q)$, but I'm still unsure because what guarantee is there that $\operatorname{vol}(\tilde K) = \operatorname{vol}(Q)$ happens (i.e. we can find such an affine image $\tilde K$)?
References:
(a) Keith Ball, An Elementary Introduction to Modern Convex Geometry.
(b) Keith Ball (1991). Volume Ratios and a Reverse Isoperimetric Inequality. Journal of the London Mathematical Society, s2-44(2), 351–359.