METRIC $X_{p}$ INEQUALITIES. (2nd February 2016)

• Record Type: Journal Article
• Authors: NAOR, ASSAF; SCHECHTMAN, GIDEON
• Is Part Of: Forum of mathematics. Volume 4(2016)
• Publication Date: 2016-02-02
• Subjects: 46B80, -- 46B85 (primary), -- 46B25, -- 47B10 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1017/fmp.2016.1
• Languages: English
• ISSNs: 2050-5086

Abstract

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.