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METRIC
$X_{p}$ INEQUALITIES
Published online by Cambridge University Press: 02 February 2016
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.
- Type
- Research Article
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s) 2016
References
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