On directions of strictness, affinity and constancy and the minimum set of a proper convex function

Abstract

The common definition of directions of affinity resp. constancy of a proper convex function implicitly requires the domain of finiteness to be unbounded in these directions. This assumption is dropped here, and directions of strictness are also defined. Under a mild continuity condition it is shown that directions of affinity resp. constancy form vector spaces. Behaviour under addition and linear transformations is investigated. Particularly simple results are obtained under the assumption that any direction is either a direction of strictness or affinity. These results are used to obtain a theorem on the dimension of the minimum set of a proper convex function, as well as simple rank conditions necessary and sufficient for strict convexity of a sum, where the summands are not necessarily strictly convex. The linear Poisson model and geometric programming are treated as examples.