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 



Find a convex hull that contains given points?
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. $c=1$.




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*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.


