Let $(X, d)$ be a compact metric space.
-
We say that $\{x_1, \cdots, x_n\} \subseteq X$ is an $\varepsilon$–covering of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, x_i) \leq \varepsilon$. Let
$$
\operatorname{Cov} (X, \varepsilon) := \min \{n: \exists \varepsilon \text {-covering of } X \text { with size } n\}
$$
be the $\varepsilon$-covering number of $X$. -
We say that $\{x_1, \cdots, x_n\} \subseteq X$ is an $\varepsilon$–packing of $X$ if $d(x_i, x_j)>\varepsilon$ for all distinct $i, j$. Let
$$
\operatorname{Pack} (X, \varepsilon) := \max \{n: \exists \varepsilon \text {-packing of } A \text { with size } n\}
$$
be the packing number of $A$.
Let $(X, d)$ and $(X', d')$ be metric spaces. The spaces $X$ and $X'$ are said to be isometric (denoted by $X \cong X'$) if there is a bijective isometry between them.
@Noam gave below example in his answer:
For $\delta \in (0, 2]$, let $E_\delta$ be the metric space consisting of three points $A,B,C$ with $d(A,B) = d(A,C) = 1$ and $d(B,C) = \delta$. Then for all $\delta, \delta' \in [1, 2]$,
$$
\operatorname{Cov} (E_\delta, \varepsilon) = \operatorname{Cov} (E_{\delta'}, \varepsilon) =
\begin{cases}
3 & \text{if} \quad \varepsilon < 1, \\
1 & \text{if} \quad \varepsilon \ge 1.
\end{cases}
$$
This example shows that $\operatorname{Cov} (X, \varepsilon) = \operatorname{Cov} (X', \varepsilon)$ for all $\varepsilon>0$ does not necessarily imply $X \cong X'$. Back to @Noam example, it's clear that
$$
\operatorname{Pack} (E_1, 1) = 0 \neq 2= \operatorname{Pack} (E_{2}, 1).
$$
I would like to ask if below statement is true, i.e.,
If $\operatorname{Cov} (X, \varepsilon) = \operatorname{Cov} (X', \varepsilon)$ and $\operatorname{Pack} (X, \varepsilon) = \operatorname{Pack} (X', \varepsilon)$ for all $\varepsilon>0$, then $X \cong X'$.
Thank you so much for your elaboration!
Best Answer
Certainly no. Consider metric spaces on $n$ points and all distance 1 and 2. There are $2^{n^2/2+o(n^2)}$ such spaces. But only polynomially many different covering and packing functions.