Timeline for Characterization of curves contained in the boundary of convex bodies
Current License: CC BY-SA 4.0
9 events
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| Oct 27, 2022 at 15:06 | history | edited | user44143 | CC BY-SA 4.0 |
clarified one sentence as per comments, moved a parenthetical definition to a less distracting place
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| Oct 27, 2022 at 11:31 | comment | added | Vadim Semenov | @TracyHall Seems very nice. I guess, this precisely means that $\gamma \subset\partial\langle\gamma\rangle$. And we can continue from there to construct a polytope. I also was interested if this polytope contains zero, but I guess your algorithm also answers this questions as well. It would be interesting to generalize this to continuous curves. I guess one will have to make a choice which projection to take at a point of a curve... Some section of tangent bundle (in a metaphorical sense, since strictly speaking it is not defined)... I will think about it. Thank you. | |
| Oct 27, 2022 at 11:21 | comment | added | Vadim Semenov | @MattF. Yes, sorry. In general, I don't know anything about this, and never thought about questions of this type. The standard convex geometry literature and the papers I am familiar with don't pursue this direction. This questions is vary vague on purpose because, in general I am interested in all possible results with similar perspective. Are there any special properties of the curves which lie on the boundary of convex bodies? What about properties of manifolds embedded in the boundary of convex bodies? A lot can be asked. | |
| Oct 27, 2022 at 4:10 | comment | added | user44143 | Will you clarify the sentence “I am looking for the reference to this characterization and related equivalent results”? Do you mean “I am looking for a reference to characterizations of these curves, or related results”? | |
| Oct 26, 2022 at 23:53 | comment | added | Tracy Hall | I don't know of any references, but in the piecewise linear case your second approach should yield an algorithm: Iterate through the line segments $\ell_i$ of $\gamma$, each of which is collapsed by a unique orthogonal projection $P_i$ to $\mathbb{R}^{n - 1}$. If for some segment $\ell_i$ the common projection of its endpoints lies in the interior of the convex hull of the projections of all other vertices of $\gamma$, then no such $K$ exists. Otherwise, it does. | |
| Oct 26, 2022 at 19:57 | history | edited | Jim Conant | CC BY-SA 4.0 |
fixed brackets
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| Oct 26, 2022 at 3:44 | history | edited | Vadim Semenov | CC BY-SA 4.0 |
added 67 characters in body
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| S Oct 26, 2022 at 3:40 | review | First questions | |||
| Oct 26, 2022 at 6:42 | |||||
| S Oct 26, 2022 at 3:40 | history | asked | Vadim Semenov | CC BY-SA 4.0 |