Let $A$ be a real matrix. Denote $\|\cdot \|$ the $p=1$ norm (sum of absolutes of the elements).
Let $C$ be all vectors (of compatible size with $A$) whose elements are in the range $[-1,1]$
How to show that $\arg \max(\|Ax \|$) over all vectors $x$ in $C$ is a vector whose elements are all either $1$ or $-1$?