Timeline for Existence of $1$-separated and $(1-\varepsilon)$-dense set in metric spaces
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 5 at 4:58 | comment | added | Christian Remling | Actually a much simpler attempt along the same lines is to take a circle of circumference $L<2$, $L\simeq 2$. Since we can put only one point on such a circle, the $\epsilon$ will be very small. It's not clear to me though how to build a space that contains circles with $L$ arbitrarily close to $2$ without the connecting pieces ruining the effect. (Also, such a space would never be uniquely geodesic.) | |
| Mar 4 at 10:01 | comment | added | Pietro Majer | 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 9:48 | comment | added | Pietro Majer | @ChristianRemling Maybe the standard symplex $\Delta_d$ (normalized with unit length edges) is easier than the $d$-cube to work with. Isn't a maximal $1$-separated subset with more than one point necessarily the $0$-skeleton? | |
| Mar 3 at 17:25 | comment | added | Christian | @AlekseiKulikov Thank you for the comment! I added the definition. | |
| Mar 3 at 17:25 | history | edited | Christian | CC BY-SA 4.0 |
Added the definition of (uniquely) geodesic metric space.
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| Mar 1 at 16:22 | comment | added | Aleksei Kulikov | Perhaps also the definition of the (uniquely) geodesic metric space could be provided? | |
| Mar 1 at 15:09 | comment | added | Christian | @an_ordinary_mathematician done. | |
| Mar 1 at 15:09 | history | edited | Christian | CC BY-SA 4.0 |
Explained the terms in the question
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| Mar 1 at 15:01 | comment | added | an_ordinary_mathematician | could you give the definition of $1$-separated and $(1-\varepsilon)$-dense ? | |
| Mar 1 at 13:44 | history | asked | Christian | CC BY-SA 4.0 |