Timeline for volume of the region above a simplex in a spherical cap
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 24, 2017 at 16:24 | comment | added | user3816 | Hi Andrew, thank you for the comment. It is not clear to me whether or not that ratio can be improved: For small $h>0$, perhaps the volume of the region is a "large enough" proportion of volume of the cap, and the volume of the base is just that of the simplex... | |
| May 24, 2017 at 15:29 | history | edited | user3816 | CC BY-SA 3.0 |
deleted 69 characters in body
|
| May 24, 2017 at 13:27 | comment | added | Andrew D. Hwang | Forgetting the simplex for the moment, cutting the unit ball by the hyperplane $x_{n} = 1 - h$ for small positive $h$ (getting a "contact lens over a small $(n-1)$-ball") gives an asymptotic volume-to-base ratio of $\frac{2h}{n+1}$. That doesn't seem definitive on its own, but it increases my skepticism that there's a "$1/n$-independent" lower bound. | |
| May 24, 2017 at 12:51 | history | edited | user3816 | CC BY-SA 3.0 |
added 8 characters in body
|
| May 24, 2017 at 12:46 | comment | added | user3816 | Hi Andrew, thank you for the comment. Correct, there is no prism with base S contained in the region, so that was not a good choice of word on my part. What I was thinking, and have updated the OP to reflect, is that perhaps one could use a sequence of sets contained in the region, whose volume, when added up, is "1/n-independent". | |
| May 24, 2017 at 12:44 | history | edited | user3816 | CC BY-SA 3.0 |
deleted 40 characters in body
|
| May 24, 2017 at 11:26 | comment | added | Andrew D. Hwang | Is this right: w/o l.o.g, your hyperplane has equation $x_{n} = h \geq 0$, each vertex of $S$ sits over a point at distance $\sqrt{1 - h^{2}}$ from $0 \in \mathbf{R}^{n-1}$, and you want (a lower bound on) the $n$-volume$$\int_{S} \bigl[\sqrt{1 - x_{1}^{2} - x_{2}^{2} - \cdots - x_{n-1}^{2}} - h\bigr]\, dx_{1} \cdots dx_{n-1}$$in terms of the $(n-1)$-volume of $S$? (If so, it appears there's no "$1/n$-independent" lower bound: Your solid doesn't contain a prism over $S$: Its height is $0$ at the vertices of $S$.) | |
| May 24, 2017 at 11:03 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Included the referenced image.
|
| May 24, 2017 at 3:15 | review | First posts | |||
| May 24, 2017 at 3:23 | |||||
| May 24, 2017 at 3:09 | history | asked | user3816 | CC BY-SA 3.0 |