Timeline for How to know if convex-hull of a set contains zero?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Aug 12, 2017 at 11:03 | comment | added | Emil Jeřábek | The property I mentioned is a trivial consequence of the fact that covnex combination preserves linear equations (and inequalitites): the constraint here is just $x_1=x_2=\dots=x_{\lambda_1}$, which is $\lambda_1-1$ linear equations. | |
| Aug 12, 2017 at 10:39 | comment | added | SMD | @EmilJeřábek Could you please give me a reference for the property you mentioned? Can we extend that property when we have $k$ different non-zero vectors $\alpha \in \mathbb{Z}^d$ (similar to the one I asked here)? | |
| Aug 11, 2017 at 18:33 | vote | accept | SMD | ||
| Aug 11, 2017 at 18:31 | comment | added | Yoav Kallus | Well, with this last added detail, you should be able to see that $\alpha_d<0$ always, so the convex hull cannot contain 0. | |
| Aug 11, 2017 at 18:28 | answer | added | Yoav Kallus | timeline score: 4 | |
| Aug 11, 2017 at 18:26 | comment | added | SMD | I am sorry guys! Added one more edit. In fact, we know that $\alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_d$. | |
| Aug 11, 2017 at 18:25 | history | edited | SMD | CC BY-SA 3.0 |
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| Aug 11, 2017 at 18:14 | comment | added | Yoav Kallus | OK, the new formulation parses, but also seems like it must be wrong. $0\in\mathbb{Z}^k$ is a point of the set, so obviously it is in its convex hull. | |
| Aug 11, 2017 at 18:14 | comment | added | Emil Jeřábek | Convex combination preserves the property that the first $\lambda_1$ coordinates are the same, etc. That is, if all the $\lambda_i$ are positive, the question is equivalent to simply if $0\in\mathrm{Conv}\{(\alpha_1,\dots,\alpha_d)\in\mathbb Z^d:\sum_{i=1}^d\alpha_i=0\}$. Either way, the answer is trivially yes, as $0$ (by which I assume you mean $(0,\dots,0)$) is already in the given set. Or am I missing something? | |
| Aug 11, 2017 at 17:57 | comment | added | SMD | Sorry for that! I rewrote it and hopefully it is clear now. | |
| Aug 11, 2017 at 17:56 | history | edited | SMD | CC BY-SA 3.0 |
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| Aug 11, 2017 at 17:44 | answer | added | Robert Israel | timeline score: 2 | |
| Aug 11, 2017 at 17:33 | comment | added | Yoav Kallus | I don't understand what the set is of which you're taking the convex hull. The way it's written now it seems like the set is $\{\lambda_i\alpha_i:i=1,\ldots,d\}$ which is a set of integers, not a set of points in $\mathbb{Z}^d$. | |
| Aug 11, 2017 at 17:07 | history | asked | SMD | CC BY-SA 3.0 |