result, using a discrete subset of $\ell_1\times\ell_1$, witnessing the same phenomena. Recently Procházka [@Pr Theorem 3] formulated an interesting metric description of reflexivity. He constructed a unifomly discrete metric space $M_R$ with the following properties: If $M_R$ bi-Lipschitzly embeds into a Banach space $X$ with distortion less than $2$, then $X$ is non reflexive. The [*distortion*]{} of a bi-Lipschitz embedding $f$ of one metric into another is the product of the Lipschitz constant of $f$ and the Lipschitz constant of $f^{-1}$. Conversely, if $X$ is non reflexive, then there exists a renorming $|\cdot|$ of $X$, so that $M_R$ embeds into $(X,|\cdot|)$ isometrically.

Our paper has the goal to find a metric characterization of reflexivity. An optimal result would be a statement, similar to Bourgain’s result, of the form “all members of a certain family $(M_i)_{i\in I}$ of metric spaces embed uniformly bi-Lipschitzly into a space $X$ if and only if $X$ is not reflexive”. In the language, introduced by Ostrovskii [@OS2], this would mean that $(M_i)_{i\in I}$ [*is a family of test spaces for reflexivity*]{}. Instead, our result will be of the form (see Theorem A below),“there is a family of sets $(M_i)_{i\in I}$, and for $i\in I$, there are 