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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?

  1. $\rho(x,y)\geq 1$ for all $x,y\in S$ with $x\neq y$
  2. 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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  • $\begingroup$ could you give the definition of $1$-separated and $(1-\varepsilon)$-dense ? $\endgroup$
    – an_ordinary_mathematician
    Commented Mar 1 at 15:01
  • $\begingroup$ @an_ordinary_mathematician done. $\endgroup$
    – Christian
    Commented Mar 1 at 15:09
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    $\begingroup$ Perhaps also the definition of the (uniquely) geodesic metric space could be provided? $\endgroup$
    – Aleksei Kulikov
    Commented Mar 1 at 16:22
  • $\begingroup$ @AlekseiKulikov Thank you for the comment! I added the definition. $\endgroup$
    – Christian
    Commented Mar 3 at 17:25
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    $\begingroup$ Uh but for the normalized $d$-simplex the best $\epsilon$ is I think $1-\sqrt{\frac d{2(d+1)}}$ (realized for $x=$ the baricenter), not o(1) $\endgroup$
    – Pietro Majer
    Commented Mar 4 at 10:01

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