A geometric Gordan-Stiemke theorem
Let C and K be closed cones in \(R^ n\). Denote by \(\phi\) (\(K\cap C)\) the face of C generated by \(K\cap C\), and by \(\phi (-K^*\cap C^*)\) the face of \(C^*\) generated by \(-K^*\cap C^*\). Then \((i)\quad\phi (K\cap C)\subseteq\phi (-K^*\cap C^*)^ D,\) where the equality holds iff \(C\cap [span(-K^*\cap C^*)]^{\perp}\subseteq K-C;\) and \((ii)\quad\phi (-K^*\cap C^*)\subseteq\phi (K\cap C)^ D,\) where the equality holds iff \(-C^*\cap [span(K\cap C)]^{\perp}\subseteq C^*+K^*.\) Hence a generalization of the Gordan-Stiemke theorem follows.
scientific article; zbMATH DE number 3871563
scientific article