Metric X_p inequalities

For every we associate to every metric space a numerical invariant such that if <![CDATA[X_p(X) and a metric space admits a bi-Lipschitz embedding into then also <![CDATA[X_p(Y). We prove that if satisfy <![CDATA[q then <![CDATA[X_p(L_p) yet. Thus, our new bi-Lipschitz invariant certifies that does not admit a bi-Lipschitz embedding into when <![CDATA[$2<q<p. This completes the long-standing search for bi-Lipschitz invariants that serve as an obstruction to the embeddability of spaces into each other, the previously understood cases of which were metric notions of type and cotype, which however fail to certify the nonembeddability of into when <![CDATA[2<q<p. Among the consequences of our results are new quantitative restrictions on the bi-Lipschitz embeddability into of snowflakes of and integer grids in, for <![CDATA[2<q<p. As a byproduct of our investigations, we also obtain results on the geometry of the Schatten trace class that are new even in the linear setting.