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.

中文翻译：

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度量不等式

对于每一个$p\in (0,\infty)$我们关联到每个度量空间$(X,d_{X})$数值不变量$\mathfrak{X}_{p}(X)\in [0,\infty]$这样如果$\mathfrak{X}_{p}(X)<\infty$和一个度量空间$(Y,d_{Y})$承认 bi-Lipschitz 嵌入到$X$然后也$\mathfrak{X}_{p}(Y)<\infty$. 我们证明如果$p,q\in (2,\infty )$满足$q<p$然后$\mathfrak{X}_{p}(L_{p})<\infty$然而$\mathfrak{X}_{p}(L_{q})=\infty$. 因此，我们新的双 Lipschitz 不变量证明了$L_{q}$不承认双 Lipschitz 嵌入$L_{p}$什么时候$2<q<p<\infty$. 这完成了长期以来对阻碍$L_{p}$彼此之间的空间，以前理解的情况是类型和共类型的度量概念，但是不能证明不可嵌入$L_{q}$进入$L_{p}$什么时候$2<q<p<\infty$. 我们的结果的后果之一是对 bi-Lipschitz 嵌入性的新定量限制$L_{p}$的雪花$L_{q}$和整数网格$\ell _{q}^{n}$， 为了$2<q<p<\infty$. 作为我们调查的副产品，我们还获得了 Schatten 几何形状的结果$p$跟踪类$S_{p}$即使在线性设置中也是新的。