Find a convex hull that contains given points? Suppose that you have vectors $a_{1},...,a_{m}$ in $\mathbb{R}^n$. Can you take $n$ among them, let $v_{1},...,v_{n}$ such that $a_{1},...,a_{m}$ belong to the convex hull of $(cn)v_{1},-(cn)v_{1},...,(cn)v_{n},-(cn)v_{n}$ for some constant $c$?
My idea was to use gram-schmidt process where at step $i$ I choose the vector $u_{i}$ with the maximum euclidean norm. The answer would be those $a_{i}$'s that maximize the at each step the euclidean norm. Any ideas? Thanks 
 A: Take $v_1,v_2,\dots,v_n$ which span a parallelepiped of maximal volume.
If 
$$a_i=x_1\cdot v_1+\dots+x_n\cdot v_n$$
then $|x_k|\le 1$, otherwise exchanging $v_k$ to $a_i$ will increase the volume.
Hence $a_i$ belongs to  the convex hull of $\{\pm n\cdot v_{i}\}$; i.e. $c=1$.

Below is the original answer to the original question.
If you agree to choose $N=\tfrac{n{\cdot}(n+1)}{2}$ points then you can get $c=1$.
Take the ellipsoid of smallest volume which contains all $\{a_i\}$.
You may assume that $\{a_i\}$ is in generic position, 
in this case at most $N$ of the points lie on the boundary of ellipsoid;
take them as $\{v_i\}$.
If one of $a_i$ does not lie in the convex hull of $\{\pm\sqrt{n}\cdot v_i\}$
then you can decrease the volume of the ellipsoid, by pushing it in one direction and expanding in all the orthogonal directions.
A: 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.  
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 (:-).
A: This is just to say that I don't think choosing the vectors with the maximum Euclidean norm will
suffice.  In the example below, the two longest vectors (red) are collinear, and the hull derived from
them—regardless of $c$—will be a line segment, which cannot contain the original (green) points.

          


