A quadratic mapping of Banach space into \(\mathbb R^ n\times \mathbb R^ n\) is \(Q(x)=A(x,x)\), where \(A: X\times X\to \mathbb R^ n\) and \(A\) is symmetric \(A(x_ 1,x_ 2)=A(x_ 2,x_ 1)\). Some topological and geometrical properties of \(H=\{x\in X, Q(x)\in C\}\), where \(C\) is a closed convex cone in \(\mathbb R^ n\), are considered. These properties are concerned with extremal problems in Banach spaces.