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.)