Remark 13.4.5 (Convex Body and Norm).label If $\norm{\cd}{}:\R^{n}\to[0,\infty)$ is a norm then $K\define \curl{x\in\R^n:\norm{x}{}\leq c}$ for any constant $c$ is a convex body. Conversely if $K$ is a convex body in $\R^{n}$ then $K=-K$ and we can define a norm $\norm{\cd}{K}$ on $\R^{n}$ to which $K$ is the unit ball.

Any norm on $\R^{n}$ is within $\sqrt{n}$ in Banach Mazur distance with the Euclidean norm. In some sense, up to a linear transformation $T$, $\frac{1}{\sqrt{n}}\norm{x}{*}\leq \norm{Tx}{2}\leq \sqrt{n}\norm{x}{*}$.

Mahler’s conjecture: for all $K\suf \R^{n}$ convex body then

$T_{n}$ is simplex. or $P_{n}$ parallelipiped instead of $T_{n}$ with $K=-K$.

Blaschke-Satalo’s: If $K=-K$

$B$ is the unit ball. If $K\neq -K$ then we take $K^{s(K)}$ instead of $K^{o}$ so we take polar w.r.t. Santalo’s point defined to be the minimum of $p\mto Vol_{n}(K^{p})$ for $p\in K$