Matrix criteria for the pseudo-P-convexity of a quadratic form (Q913906)

A function f(x,y) of vector variables x and y is said to the pseudo-P- convex if the inequalities \((x_ 2-x_ 1)^ T\nabla_ xf(x_ 1,y_ 1)\geq 0\) and \((y_ 2-y_ 1)^ T\nabla_ yf(x_ 1,y_ 1)\geq 0\) imply \(f(x_ 2,y_ 2)\geq f(x_ 1,y_ 1)\) for all \((x_ 1,y_ 1),(x_ 2,y_ 2)\in b_ 1\times b_ 2,\) some open convex set in \(R^ n\times R^ m\). The authors characterise quadratic functions f which are pseudo- P-convex on a solid convex subset of \(R^ n\times R^ m\). It is shown that such quadratic forms have at most two negative eigenvalues. The authors characterise a subset \({\mathcal D}\) of \(R^ n\times R^ m\) for a given quadratic form f such that f is pseudo-P-convex on \(b_ 1\times b_ 2\) if and only if \(b_ 1\times b_ 2\subset {\mathcal D}\). It is shown that \({\mathcal D}\), though not convex, however contains three pairs of disjoint convex cones, which are characterised by quadratic forms.