# Existence of $1$-separated and $(1-\varepsilon)$-dense set in metric spaces

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.

• could you give the definition of $1$-separated and $(1-\varepsilon)$-dense ? Mar 1 at 15:01
• @an_ordinary_mathematician done. Mar 1 at 15:09
• Perhaps also the definition of the (uniquely) geodesic metric space could be provided? Mar 1 at 16:22
• @AlekseiKulikov Thank you for the comment! I added the definition. Mar 3 at 17:25
• 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) Mar 4 at 10:01