An elementary exclusion principle for Michell trusses (Q694695)

In this very important and useful paper, the author proves a result which implies the following for optimal structures: if a cable lies between two bars (or a bar between two cables) and they share an endpoint, then the angle between the two bars (respectively, cables) must be 180 degrees. (It is known that at points where strain function is smooth, bars and cables can only intersect orthogonally). The main result reads as follows: For \(F = \sum ^3_{l=1} \lambda_l (\delta_{a_j} - \delta_a) \widehat{a_j - a}\) where \(\lambda_1, \lambda_2 >0, \lambda_3 <0, a_1-a \) and \(a_2-a\) are linearly independent, and \(a_3-a= \alpha (a_1-a) + \beta (a_2 - a)\) for \(\alpha, \beta >0\), there exists a decomposition of \(F\) with strictly smaller cost. (The author presents a sufficient condition for the non-optimality of a given decomposition, even if the decomposition is infinite.) If \(a_{i_0} \in \mathcal{A}\) and \((\Lambda, \mathcal{A})\) is optimal then do not exist \(i_1, i_2, i_3 \in N\) such that \(a_{i_3} - a_{i_0} = \lambda ( a_{i_1}-a_{i_0})+ \beta (a_{i_2} - a_{i_0})\) for linearly independent \(a_{i_1} - a_{i_0}\) and \(a_{i_2}-a_{i_0}\) for some \(\alpha, \beta >0\) and either \(\lambda_{i_1, i_0}, \lambda_{i_2, i_0} >0, \lambda_{i_3, i_0} <0\) or \(\lambda_{i_1, i_0}, \lambda_{i_2, i_0} <0, \lambda_{i_3, i_0} >0\) (for an equilibrated force \(F= \sum^n_{i=1}f_i \delta_{m_i}\) and \(\Lambda = \{ \lambda_{i,j}\}^\infty _{i,j=1} \subset R\) and \(\mathcal{A} = \{ a_i\}^\infty _{i=1} \) satisfy \(\text{Cost} (\Lambda, \mathcal{A}) = \sum^\infty_{i,j=1} |\lambda_{i,j}| \quad \|a_i-a_j\| < \infty\) and \(F= \sum^\infty_{i,j=1} \lambda_{i,j}(\delta_{a_i} - \delta_{a_j}) \widehat{a_i - a_j}\) in the weak sense).