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Convex polyhedron $P$ is a subset of $\mathbb{R}^n$ that satisfies system of linear inequalities \begin{align} a_{11}x_1 + \cdots + a_{1n}x_n & \sim_1\, c_1 \\ & \vdots \\ a_{p1}x_1 + \cdots + a_{pn}x_n & \sim_p\, c_p, \end{align} where $\sim_i \in \{\leq,\geq\}$. It can be alternatively represented by two finite sets of generators $V, W \subseteq \mathbb{R}^n$: $$P = \text{conv}(V) + \text{cone}(W),$$ where conv(V) denotes all convex combinations of points in V and cone(W) all nonnegative linear combinations of points in W.

Now, what if we allow $\sim_i$ to be from $\{\geq,>,\leq,<\}$. Is there some similar representation in terms of generating points for such sets?

(I possess no knowledge of this area of mathematics, so I apologize if I got the terminology wrong or if this question is just plain stupid.)

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    I may misunderstand the question because it seems trivially false -- take 02010-10-08
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    I didn't mean that the set must neccessarily be defined as P = conv(V) + cone(W) for some V and W. What I wanted to know is if the set can be described /somehow/ similarly, in terms of generating points. I am aware that it is rather vague question, but there is always chance that it may be something obvious or something almost everyone knows. Anyway, my mistake, I'll try to rephrase the question.2010-10-08

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