On an invariant related to a linear inequality

Abstract

Let$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } = (\alpha _1 ,\alpha _2 , \ldots \alpha _m ) \in \mathbb{R}_{ > 0}^m$. Let$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } _{i,j}$ be the vector obtained from$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha }$ on deleting the entries α_i and α_j. We investigate some invariants and near invariants related to the solutions$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \in } \in \{ \pm 1\} ^{m - 2}$ of the linear inequality

$$\left| {\alpha _i - \alpha _j } \right|< \left\langle {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \in } ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } _{i,j} } \right\rangle< \alpha _i + \alpha _j ,$$

, 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.