On an invariant related to a linear inequality (Q1865725)

For a sequence \((\alpha_1 ,\alpha_2, \ldots ,\alpha_m)\in \mathbb{R}^m\) and given \(1\leq i<j\leq m\) the authors consider a signed count of all solutions of \[ |\alpha_i -\alpha_j |< \sum \varepsilon_k \alpha_k < \alpha_i+\alpha_j \] where \(\varepsilon_i=\varepsilon_j=0\) and all other \(\varepsilon =\pm 1\). For \(m\) odd 3 proofs are given that this signed count is independent of the choice of \(i,j\). Several related results are proved and this invariant is related to Rademacher functions.