Is it know which metric spaces $M$ do have the following property: there is $\varepsilon>0$ and a maximal $1$-separated set which is $(1-\varepsilon)$-dense?

In other words, when does at set $S\subset M$ with the following properties exits?

- $\rho(x,y)\geq 1$ for all $x,y\in S$ with $x\neq y$
- For every $x\in M$ there is a $y\in S$ with $\rho(x,y)\leq 1-\varepsilon$.

In particular, we would be interested whether locally compact (uniquely) geodesic metric spaces have this property.

A metric space is geodesic, if for every pair of points $x,y\in M$ there is an isometry $\gamma\colon [0,\rho(x,y)]\to M$ with $\gamma(0)=x$ and $\gamma(\rho(x,y))=y$. It is called uniquely geodesic if $\gamma$ is unique.

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