name for a polytope constructed from a system of linear equations? To a system of inhomogeneous linear equations, one can associate a polytope, as follows. Let $A\in\mathbb{R}^{m\times n}$, 
$b\in\mathbb{R}^m$ and $V=\{x\mid Ax=b, \text{support of $x$ minimal}\}\subset \mathbb{R}^n$, where by the support of a vector I mean the index set of nonzero entries. Assume $V$ finite (this is not always the case, but happens often enough).
Now the polytope $P(A,b)$ in question is the convex closure of $V$.
How does one call $P(A,b)$? Were such things studied anywhere?
Edit: a way to deal with it might be to blow up the dimension and work instead
with $Y:=\{y\geq 0 \mid (A; -A)y=b\}\subset\mathbb{R}^{2n}$. Then $Y=S+C$, with $S$ a polytope and $C$ the recession cone of $Y$; the vertices of $S$ are naturally split into
pairs $(x_+,x_-)$,  with $x_+,x_-\in\mathbb{R}^n$, and $x_+$ and $x_-$ having disjoint support (provided $b\neq 0$). Thus $A(x_+-x_-)=b$, and it looks as if it should hold that each $x\in V$ gives rise to a vertex $(x_+,x_-)$ of $S$, by splitting $x$ into positive and negative parts.
 A: Maybe a term you are looking for is "sparse solutions of underdetermined linear systems". At least this would maybe be a good name for your set $V$. It can be viewed as solution set of the following optimization problem:
$$\min||x||_0 \text{ subject to }Ax=b.$$
where $||\cdot||_0$ counts the number of nonzeros.
Then $P(A,b)$ could be called "convex hull of sparse solutions".
References in this direction are for example Donoho and a survey by Lai.
A: Dima, your edit is correct. Note that the generators of the recession cone are in 1-1 correspondence to the circuit of the matrix $A$ (just split them into positive and negative parts), plus the obvious generators $(e_i,e_i)$.
Finding (overall) support minimal nonnegative solutions to a linear system of equations is an NP-hard problem in optimization, but I am not aware of consideration of the set of all such sparse solutions (including those with a non-minimum number of nonzeros). A natural suggestion for your polytope would be "polytope of sparse solutions".
Do you really have examples with infinite sets V? This looks impossible to me (= I could provide a proof).
Do you have some paper (by now) where you deal with this polytope? I'd be happy to read it.
