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This is a follow-up to this post.

Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by $$ \kappa_X(\epsilon)\triangleq\sup\left\{ k : \exists x_0,\dots,x_k \in X : \exists r>0, \sqcup_{i=1}^k B(x_i;\epsilon r)\subseteq B(x_0;r) \right\} . $$

If $(\mathbb{R}^D,d_{Euc})$ is a Euclidean space, with $D$ large enough, such that there exists a $C>0$ satisfying $$ K_X(\epsilon) \leq CK_{\mathbb{R}^D}(\epsilon);\qquad \left(\forall \epsilon \in \Big[0,\frac1{3}\Big)\right). $$

Does there necessarily exist a continuous injection of $(X,d_X)$ into a metric ball in $(\mathbb{R}^D,d_{Euc})$ of some (finite) radius $M>0$?

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  • $\begingroup$ A typo... now it's Euclidean space. $\endgroup$
    – ABIM
    Commented Jan 21, 2019 at 9:29
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    $\begingroup$ It is well-known that a compact metric space $X$ admits a continuous injection into some Euclidean space if and only if $X$ has finite topological dimension. More precesely, each metrizable separable space $X$ of finite dimension $n$ admits a topological embedding to the Euclidean space $\mathbb R^{2n+1}$. $\endgroup$
    – Taras Banakh
    Commented Jan 21, 2019 at 11:33

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