Timeline for Asymptotic cone
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 29, 2021 at 7:48 | vote | accept | Hebe | ||
| Aug 28, 2021 at 9:25 | comment | added | YCor | However the sequence of subset $(1/n)S$ might not converge (think of $S=\bigcup_n [2^{2^n},n2^{2^n}]$ in $\mathbf{R}$). The definition I'm aware is that asymptotic cones are all limit points of this sequence (they exist by compactness). The definition you use is rather some kind of limsup (possibly the closure of the union of all asymptotic cones, in my sense). When it converges these coincide. | |
| Aug 28, 2021 at 8:42 | comment | added | Hebe | @YCor I see. Thank you! This is indeed a good way to imagine the asymptotic cone. | |
| Aug 28, 2021 at 8:16 | comment | added | YCor | I mean: if when $r\to 0$, the subsets $rS$ converge (say in Hausdorff topology, locally), the asymptotic cone is the limit. Since rescaling $S$ as $rS$ with small $r$ is like looking $S$ from far away, you see the asymptotic cone by looking $S$ from far away as the limit shape. (And the asymptotic cone is always closed.) | |
| Aug 28, 2021 at 5:16 | comment | added | Hebe | @YCor Sorry, I do not quite understand what you mean. Could you please make it more explicit? | |
| Aug 27, 2021 at 17:22 | answer | added | Iosif Pinelis | timeline score: 1 | |
| Aug 27, 2021 at 16:18 | history | edited | YCor |
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| Aug 27, 2021 at 16:17 | comment | added | YCor | Just draw the picture of your set with a far away scale (namely draw $S\cap [-100,100]^2$ in a square) and you have a good representation of what is the asymptotic cone. | |
| Aug 27, 2021 at 15:58 | history | asked | Hebe | CC BY-SA 4.0 |