Separation via quadratic functions

Summary

The subject of this paper is to study the separability of two given functions by a quadratic function.

LetG be an abelian group. A functionQ: G → [− ∞, ∞[is calledquadratic if

$$Q(x + y) + Q(x - y) = 2[Q(x) + Q(y)].$$

LetP, R: G → [− ∞, ∞[ be given functions withP ≤ Q onG. Theseparability ofP andR by a quadratic function means that there exists a quadratic functionQ satisfyingP ≤ Q ≤ R onG.

The following problem and its motivation comes from the theory of second-order directional derivatives: Given a functionP: G → [− ∞, ∞[, we look for the existence of a family

of quadratic functions such that

In order to be able to attack this problem, we shall need necessary and sufficient conditions on the quadratic separability of arbitrary functionsP andR.

In three sections of the paper we investigate the quadratic separability in various settings. First the case is considered when the values ofP andR are finite and the underlying structure is a group. Then we deal with functionsP andR that may take the value − ∞. However, in this case the underlying structure is assumed to be a vector space overQ. In the third section we treat the case when topological assumptions are present, and hence the separation by continuous quadratic functions is required. The paper contains an application and examples that explain the differences between the main results of the paper.