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Suppose I have a set of generators $G \in R^{n\times d}$.

Where $n>d$. Here we have $n$ generators in $d$ dimension.

And I'm only interested in the local structure of 1 vertex $\vec{p}= \sum_{i=1}^n \vec{g_i}$.

Which is all the facets $\vec{p}$ is in.

Is there any fast algorithm to do so?

For example for 3 generators in 2d. We can get the two vectors connected with $\vec{p}$ by sorting the generators by angles and grab the minimum and maximum.

And there is a vertex enumeration algorithm for zonotope, but since I'm interested only in the local structure, I'm wondering if there exists anything simplier than enumerate all the vertex

peng yu
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  • Your notations are too compact. For example, one must guess if you are considering $n$ points in $\mathbb{R}^p$ or $p$ points in $\mathbb{R}^n$. Moreover $p=1.G$ is unnecessarily cryptic. Please take an example. Otherwise, I fear that you will have no answers because people are unsure about understanding your notations. Besides, are you familiar with the algorithmic domain called "linear programming" (simplex algorithm, etc.) ? You could find an answer on this side – Jean Marie May 15 '20 at 08:32
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    thank you for your comment, i have updated the description of the problem. – peng yu May 15 '20 at 13:56
  • Thanks. I have been spending time on particular zonohedra (see Edit part in (https://math.stackexchange.com/q/2127476)) but I haven't been thinking to this issue. A possible keyword for this is "minkowski sum of line segments" like in (https://cs.gmu.edu/~jmlien/masc/uploads/Main/lien_wafr08.pdf) – Jean Marie May 15 '20 at 14:29
  • thanks! for 2d we can just sort the vectors by their angle, and get the first and last vector. I'm wondering if there is anything convenient in 3d space. I really want to avoid the linear programming stuff – peng yu May 18 '20 at 04:11

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