On an invariant related to a linear inequality

Let α = (α₁, α₂, ..., α_m) ∈ ℝ_>0^m. Let α_i,j be the vector obtained from α on deleting the entries α_i and α_j. We investigate some invariants and near invariants related to the solutions ∈ ∈ {± 1}^m-2 of the linear inequality |α_i - α_j| < 〈 ∈, α_i,j 〉 < α_i + α_j, where 〈, 〉 denotes the usual inner product. One of our methods relates, by the use of Rademacher functions, integrals involving products of trigonometric functions to these quantities.