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
So a $P(A,b)$ could be called "convex hull of sparse solutions".

References in this direction are for example [Donoho][1] and a survey by [Lai][2].


  [1]: http://statistics.stanford.edu/~ckirby/techreports/GEN/2005/2005-04.pdf
  [2]: http://ftp.math.uga.edu/~mjlai/papers/surveySS.pdf