I have a problem that seems like it should have a slick, elegant solution but I'm having trouble finding one.
I'm working with convex polytopes with vertices that are subsets of $\{-1,1\}^n$. When considering the facet-defining hyperplanes $$a_1 x_1 + \cdots + a_n x_n + b = 0$$ I'd like to claim that we may assume the coefficients $a_i$ lie in $\{0, \pm 1\}$.
I have a clunky induction argument in mind, but it feels like there may be a shrewd linear algebra trick for something like this.
Thanks!