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

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