On an invariant related to a linear inequality

Let A be an m-dimensional vector with positive real entries. Let A_{i,j} be the vector obtained from A on deleting the entries A_i and A_j. We investigate some invariant and near invariants related to the solutions E (m-2 dimensional vectors with entries either +1 or -1) of the linear inequality |A_i-A_j| < <E,A_{i,J}> < A_i+A_j, where <,> denotes the usual inner product. One of our methods relates, by the use of Rademacher functions, integrals involving trigonometric quantities to these quantities.