Timeline for Second derivative of the volume of the $\varepsilon$-neighbourhood of a submanifold
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 30, 2023 at 23:55 | vote | accept | Nate River | ||
| Jan 30, 2023 at 20:29 | answer | added | Anton Petrunin | timeline score: 3 | |
| Aug 3, 2021 at 4:29 | comment | added | Igor Belegradek | I should have mentioned that the point of Weyl's paper is that the mixed volumes are independent of the embedding, i.e. they are invariants of the metric on $N$, which is quite remarkable. See also en.wikipedia.org/wiki/Weyl%27s_tube_formula. | |
| Aug 3, 2021 at 4:23 | comment | added | Nate River | Wow that’s more complicated than expected. I would hope that taking the limit makes things somewhat easier so we might be able to avoid explicit expressions (for fixed $\varepsilon > 0$, that is). | |
| Aug 3, 2021 at 4:15 | comment | added | Igor Belegradek | You may want to check Hermann Weyl's "On the Volume of Tubes", jstor.org/stable/2371513?seq=1#metadata_info_tab_contents. If $M=\mathbb R^n$, then the volume of $N_\varepsilon$ is a degree $n$ polynomial in the variable $\varepsilon$ whose coefficients are mixed volumes of $N$ (up to constant multiples). Weyl gives an explicit formula for the mixed volumes. | |
| Aug 3, 2021 at 2:57 | history | asked | Nate River | CC BY-SA 4.0 |