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It is known that every compact (closed and bounded) polyhedron $P$ can be written as a convex hull of finitely many points, i.e., $\text{conv}\{x_1, \dots, x_m\}$.

Questions

  1. How can I make sure that, I am always able to find these $m$ points?
  2. Are these points vertices of the polyhedron?
  3. When we find these $m$ points, are they unique?
  4. Under what condition it true that $P \backslash \{x_1, \dots, x_m\}=\{\theta_1x_1+\dots+\theta_mx_m \mid\theta_i>0, \sum_i^m \theta_i=1, i=1,\dots,m\}$?

My thoughts:

Since $P$ is a polyhedron, one needs finitely many inequalities and equalities to represent it. We should be able to come up with some methods to extract vertices from them.

Saeed
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  • Is this homework? For 4. consider a triangle to see tat the answer is no. – copper.hat Jan 22 '22 at 04:41
  • @copper.hat: no it is not a homework problem. I want to figure out how I can show it rigorously. Also, I edited 4. – Saeed Jan 22 '22 at 04:45
  • How familiar are you with fundamental polytope theory? It's relatively well-known that the answers to the first three are positive. –  Jan 22 '22 at 04:56
  • If you take a (non degenerate) triangle in the plane, the right hand side above is the interior. 1. is the vertex enumeration problem. 2. Not necessarily, but they contain the vertices. 3. No, you can always add points that are already in the set. – copper.hat Jan 22 '22 at 04:57
  • @William: I am not. I am looking for an understandable proof. – Saeed Jan 22 '22 at 04:58
  • You need some minimality conditions in order to assert that the $m$ points are unique. – copper.hat Jan 22 '22 at 04:59
  • @copper.hat: can you please help me to figure it out? – Saeed Jan 22 '22 at 05:10
  • Figure what out exactly? – copper.hat Jan 22 '22 at 05:10
  • @copper.hat: I am trying to understand your solution here and I have a proof in my mind. I cannot see why you use $P\backslash {x_1, \dots,x_m}$. – Saeed Jan 22 '22 at 05:14
  • @Sepide I made an incorrect statement there. – copper.hat Jan 22 '22 at 05:22
  • @copper.hat: can you please explain that incorrect statement there to me and answer my questions under your solution there? – Saeed Jan 22 '22 at 05:24
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    As copper.hat.has pointed out, there are some minimality conditions to be had--- in particular, the first three questions are positive only if we assume $m$ is minimal. These problems are nontrivial, and a proof is not short. I recommend Ziegler for this; such problems are dealt with at the beginning of the book. –  Jan 22 '22 at 05:31
  • I like Rockafellar's Convex Analysis. To find the vertices you could look at https://or.stackexchange.com/q/4540/2547. – copper.hat Jan 22 '22 at 06:03
  • Also, $ P\setminus {x_1, \dots, x_m}={\sum_k \theta_k x_k \mid\theta_i \ge 0, \sum_k \theta_k = 1, \theta_k <1 } $. – copper.hat Jan 22 '22 at 06:06

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