Find a polytope that contains all the 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 $(c\sqrt{n})v_{1},-(c\sqrt{n})v_{1},...,(c\sqrt{n})v_{n},-(c\sqrt{n})v_{n}$ 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: I've found a counterexample to the smaller ($c\sqrt{n}$) bounds:
First, take $n\geq 3$ and $m = n+1$.  Let
$a_1 = (0, 0, \ldots, 0, n)$
$a_2 = (1, 0, \ldots, 0, 1)$
$a_3 = (0, 1, \ldots, 0, 1)$
$\vdots$
$a_n = (0, 0, \ldots, 1, 1)$
$a_m = (1, 1, \ldots, 1, 0)$
Observe that each of these vectors can be expressed uniquely as a linear combination of the others:
$a_1 = \frac{n}{n-1}(a_2 + \cdots + a_n - a_m)$
$a_2 = \frac{n-2}{n}a_1 - a_3 - a_4 - \cdots - a_n + a_m$
$a_3 = \frac{n-2}{n}a_1 - a_2 - a_4 - \cdots - a_n + a_m$
$\vdots$
$a_m = -\frac{n-1}{n}a_1 + a_2 + \cdots + a_n$
Now, observe that sums of the absolute values of these coefficients grow $O(n)$:
$\|(a_1)_{\beta_1}\|_1 = \|\frac{n}{n-1}(1, 1, \ldots, 1, -1)\|_1 = n^2/(n-1)$ 
$\|(a_2)_{\beta_2}\|_1 = \|(\frac{n-2}{n}, -1, \ldots, -1, 1)\|_1 = (n^2-2)/n$
$\|(a_3)_{\beta_3}\|_1 = \|(\frac{n-2}{n}, -1, \ldots, -1, 1)\|_1 = (n^2-2)/n$
$\vdots$
$\|(a_m)_{\beta_3}\|_1 = \|(-\frac{n-1}{n}, 1, \ldots, 1, 1)\|_1 = (n^2-1)/n$
where $\beta_i = (a_1, \ldots, a_{i-1}, a_{i+1}, \ldots, a_m)$ is the ordered basis of the vectors excluding $a_i$.  Observe that these $\ell^1$ norms give the minimum factor $k$ such that the convex hull of $k\{a_j\}_{j\neq i}$ contains $a_i$.  Since $k$ grows $O(n)$ (regardless of your choice of $\{v_j\} = \{a_j\}_{j\neq i}$), you can't find a constant $c$ such that $c\sqrt{n}\{v_j\}$ contains $\{a_i\}$, for some choice of $\{v_j\}$, for all $n$.
Found this looking for a positive proof.  I'm now trying to prove it positively for the $cn$ bounds :-)
