METRIC X_p INEQUALITIES. (2nd February 2016) - NAOR, ASSAF; SCHECHTMAN, GIDEON
Abstract : For every p(0, ∞ ) we associate to every metric space (X, d_X) a numerical invariant X_p(X)[0, ∞ ] such that if X_p(X)<∞ and a metric space (Y, d_Y) admits a bi-Lipschitz embedding into X then also X_p(Y)<∞ . We prove that if p, q(2, ∞ ) satisfy q<p then X_p(L_p)<∞ yet X_p(L_q)=∞ . Thus, our new bi-Lipschitz invariant certifies that L_q does not admit a bi-Lipschitz embedding into L_p when 2<q<p<∞ . 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<∞ . 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 _q^n, for 2<q<p<∞ . 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.