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i'm not a mathematician. So, my question may be stupid. Hope that it won't make you angry.

Is there any mathematical relationship between the vertex representation and the half-space representation of a given polytope?

Clearly speaking, i have the vertex representation of a polytope $P$: $P = conv\left\lbrace v_1, ...,v_l\right\rbrace \subset \mathbf{R}^n$. It is easy to compute its half-space representation (by the help of a software e. Fukuda's CDD). Now i would like to add a vector $a_i$ in e.g. $\mathbf{R}^m$ to each $v_i$. I will have a polytope denoted e.g $P^{aug} \subset \mathbf{R}^{m+n}$ and $P^{aug}=conv \left\lbrace \left[\begin{matrix}v_1 \\a_1 \end{matrix}\right], ..., \left[\begin{matrix}v_l \\a_l \end{matrix}\right] \right\rbrace$.

Can we say somethings about these additional vectors $a_i$, if i want the half-space representation of $P^{aug}$ have a pre-defined structure e.g $H\left[\begin{matrix}v \\a \end{matrix}\right] \leq K$, where $H, K$ are know matrices?

Thank you, hope to receive precious answers.

noone
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  • Is this a "theoretical" or "practical" question? In theory, you can easily find the corners (assuming the polytope is bounded), upon which you can check whatever you need about the vertices. –  Mar 23 '14 at 20:51
  • could you clarify more please, i would like to find a theoretical result about it. – noone Mar 23 '14 at 20:57
  • Theoretically, assuming dimension $N$, the vertices are the intersections of $N$-tuples of the hyperplanes satisfying the remaining inequalities. Of course, if you choose your $H$ and $K$ at random, it is extremely unlikely that the vertices will have the given projections $v_i$ (or even that their number will be the prescribed one). So, you should say something about $H$ and $K$ and how they are related to the original polytope. –  Mar 23 '14 at 21:12
  • yeah, i understand what you mean, for the moment, i have not had any idea about $H, K$, but if i suppose the values of $H, K$ such that there exists $a_i$, is there any study which does find such $a_i$? – noone Mar 23 '14 at 21:21
  • I don't know; it's too vague. I would start with a study of the conditions on $H$ and $K$ that would guarantee such $a_i$. Then, with probability $1$, $a_i$ would be easy to find :) –  Mar 23 '14 at 21:27
  • thank you for your advises. – noone Mar 23 '14 at 21:35
  • Another way to put it: the question "how" is meaningless without "when". If you don't want to answer the latter, just keep your fingers crossed and find the vertices: they are uniquely determined by the data. Then see if they are as you want; if they are, you have your $a_i$. –  Mar 23 '14 at 21:36

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