John's theorem and factorizations of linear operators

The correspondence between convex sets and finite dimensional Banach spaces

Let $K \subseteq \R^n$ be a symmetric compact convex set with nonempty interior. Such a convex set is called a convex body. We can associate a norm with $K$, called the gauge of $K$, as follows:

It readily follows that $\| \cdot \|_K$ is the norm such that $K$ is the unit ball of that norm. A finite dimensional Banach space is essentially $\R^n$ using some $K$ as the gauge of a norm. Further, the unit ball of any Banach space is a convex body; thus symmetric convex sets and finite dimensional Banach spaces are practically interchangeable. The prototypical example is $\ell_2^n$, whose unit ball is the regular 'spherical' ball $\{x \in \R^n \mid \sum_{i=1}^n x_i^2 \leq 1\}$. This correspondence lets us translate inequalities with applications to functional analysis to nice (at least in my opinion) geometric statements about the relations of different convex bodies to each other.

Factorizations