AI 中文总结
本文证明每个非空紧致度量空间均可等距嵌入Gromov-Hausdorff空间，通过将1-Lipschitz函数族实现为Cantor空间上的度量族，并给出有限逼近与精确距离公式。
AI 中文摘要
设$(\mathcal{M},d_{\mathrm{GH}})$表示非空紧致度量空间的等距类构成的Gromov-Hausdorff空间。我们证明每个非空紧致度量空间都可等距嵌入到$(\mathcal{M},d_{\mathrm{GH}})$中。更精确地，对于每个$D>0$和每个满足$\operatorname{diam} K\le D$的非空紧致度量空间$K$，我们将$K$上所有取值于$[0,D]$的$1$-Lipschitz函数的空间实现为固定Cantor空间上的一族度量。在此实现下，Gromov-Hausdorff距离恰好与函数之间的均匀距离一致，并且每个所得度量空间的直径至多为$76D$。我们还构造了有限逼近，其Gromov-Hausdorff距离由精确公式给出，并附带一个均匀逼近估计。
英文摘要
Let $(\mathcal{M},d_{\mathrm{GH}})$ denote the Gromov-Hausdorff space of isometry classes of nonempty compact metric spaces. We prove that every nonempty compact metric space is isometrically embeddable into $(\mathcal{M},d_{\mathrm{GH}})$. More precisely, for every $D>0$ and every nonempty compact metric space $K$ with $\operatorname{diam} K\le D$, we realize the space of all $1$-Lipschitz functions on $K$ with values in $[0,D]$ as a family of metrics on a fixed Cantor space. Under this realization, the Gromov-Hausdorff distance agrees exactly with the uniform distance between functions, and each resulting metric space has diameter at most $76D$. We also construct finite approximations for which the Gromov-Hausdorff distance is given by an exact formula, together with a uniform approximation estimate.
Comments14 pages
