Metric ${X}_p$ inequalities

For every $p\in (0,\infty)$ we associate to every metric space $(X,d_X)$ a numerical invariant $\mathfrak{X}_p(X)\in [0,\infty]$ such that if $\mathfrak{X}_p(X)<\infty$ and a metric space $(Y,d_Y)$ admits a bi-Lipschitz embedding into $X$ then also $\mathfrak{X}_p(Y)<\infty$. We prove that if $p,q\in (2,\infty)$ satisfy $q<p$ then $\mathfrak{X}_p(L_p)<\infty$ yet $\mathfrak{X}_p(L_q)=\infty$. Thus our new bi-Lipschitz invariant certifies that $L_q$ does not admit a bi-Lipschitz embedding into $L_p$ when $2<q<p<\infty$. This completes the long-standing search for bi-Lipschitz invariants that serve as an obstruction to the embeddability of $L_p$ 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 $L_q$ into $L_p$ when $2<q<p<\infty$. Among the consequences of our results are new quantitative restrictions on the bi-Lipschitz embeddability into $L_p$ of snowflakes of $L_q$ and integer grids in $\ell_q^n$, for $2<q<p<\infty$. As a byproduct of our investigations, we also obtain results on the geometry of the Schatten $p$ trace class $S_p$ that are new even in the linear setting.