Metric $X_p$ inequalities

Abstract

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.

Comments of referee addressed. To appear in Forum Math, Pi

References in corpus (5)

• Nonlinear spectral calculus and super-expanders
• Expanders with respect to Hadamard spaces and random graphs
• Spectral calculus and Lipschitz extension for barycentric metric spaces
• Improved bounds in the metric cotype inequality for Banach spaces
• Improved bounds in the scaled Enflo type inequality for Banach spaces

Cited by in corpus (1)

• Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete