Farkas' lemma and multilinear forms (Q1011840)

The author shows that the analogue of Farkas' Lemma can fail if one substitutes linear forms with bilinear ones. Specifically, he provides an example of three bilinear forms \(A, B,\) and \(C\) taking \(\mathbb R^2 \times \mathbb R^2 \to \mathbb R\) with the following property: Whenever \(A(x,y) \geq 0\) and \(B(x,y) \geq 0,\) then \(C(x,y) \geq 0,\) although \(C\) is not a linear combination of \(A\) and \(B.\) Further details and a related positive result are available in work by \textit{M.\,Maestre}, the author, and the reviewer [Note Mat.\ 25(2005--2006), No.\,1, 49--54 (2006; Zbl 1221.15034)].