John's theorem and factorizations of linear operators

August 20th, 2023

The correspondence between convex sets and finite dimensional Banach spaces

Let KRnK \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 KK, called the gauge of KK, as follows:

xK=inf{t0xtK}\|x\|_K = \inf \{t \geq 0 \mid x \in t K\}

It readily follows that K\| \cdot \|_K is the norm such that KK is the unit ball of that norm. A finite dimensional Banach space is essentially Rn\R^n using some KK 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 2n\ell_2^n, whose unit ball is the regular 'spherical' ball {xRni=1nxi21}\{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.


Theorem.(Dvoretzky-Rogers Lemma)

Let (X,X)(X, \|\cdot\|_X) be an nn-dimensional Banach space. Then there exists a set of m=nm = \sqrt{n} vectors x1,,xmRnx_1, \ldots, x_m \in \R^n such that for all t1,,tmRt_1, \ldots, t_m \in \R

max1imtii=1mtixiXC(i=1mti2)1/2\max_{1 \leq i \leq m} |t_i| \leq \left\| \sum_{i=1}^m t_i x_i \right\|_X \leq C \cdot \left(\sum_{i=1}^m t_i^2 \right)^{1/2}

where C>0C > 0 is a universal constant.