t,\,\ x\in B_X\Big\}>0,$$ for all $t>0$, where $\text{sep}\big[(z_n)\big]=\inf_{m\not=n} \|z_m-z_n\|$, for a sequence $(z_n)\subset X$. The function $\bar\beta_X$ is called the [*$\beta$-modulus of $X$*]{}. Using the equivalence between the positivity of the $\beta$-modulus and the property that a separable Banach space is reflexive and (AUC)- and (AUS)-renormable, Baudier [@Ba3] was able to establish a new and shorter proof of the above cited result from [@BKL]. Metric descriptions of other non-local Banach space properties, for example the Radon-Nikodým Property, can be found in [@Os3].

In our paper we would like to concentrate on metric descriptions of the property that a Banach space is reflexive, and subclasses of reflexive Banach spaces. In [@Os3], Ostrovskii established a [*submetric*]{} characterization of reflexivity. Let $T$ be the set of all pairs $(x,y)$ in $\ell_1\times \ell_1$, for which $\|x-y\|_{1}\le 2 \|x-y\|_s$, where $\|\cdot\|_1$ denotes the usual norm on $\ell_1$ and $\|\cdot\|_s$ denotes the [*summing norm*]{}, [*i.e.,* ]{} $\|z\|_s=\sup_{k\in{\mathbb{N}}}\big|\sum_{j=1}^k z_j\big|$, for $z=(z_j)\in \ell_1$. [@Os3 Theorem 3.1] states that a Banach space $X$ is not reflexive if and only if there is a map $f:\ell_1\to X$ and a number $0<c\le 1$, so that $c\|x-y\|_1\le \| f(x)-f(y)\|\le \|x-y\|_1$ for all $(x,y)\in T$. In Section \[S:13\] we will formulate a similar 