This reduces to the problem of finding a set of linearly independent vectors with maximum cardinality.  There are in general many such sets, but any of them is a solution if you pick $c$ large enough.  Then the convex hull is an $n$-orthoplex (AKA cross-polytope).  If you make $c$ big, it will include any set of points in the span of the set, including on particular $a_1,...,a_m$.  Some googling reveals that an algorithm for finding such a set is [here][1].  

Or are you trying to produce a solution with minimum $c$?  This is a much more interesting question, by which I mean that I don't know the answer (:-).

  [1]: http://math.stackexchange.com/questions/164016/how-to-find-maximal-linearly-independent-subsets