Let $a, b, c,$ and $d$ each be variable and bounded between $0$ and $1$, inclusive, and be such that $a+b+c+d=1$. Further, let $α, β, γ, δ,$ and $ε$ be known constants. Then for what fraction of the space of values that $a,b,c,$ and $d$ can take is the following inequality satisfied?
$\pmatrix{a & b& c& d} \pmatrix{α \\ β \\ γ \\ δ} > ε$
My knowledge of linear algebra is obviously not very deep, but my intuition is that the problem can be solved through a series of linear transformations. I am willing to do a lot of reading if someone can point me in the right direction. Thanks!