Gerstewitz Functionals on Linear Spaces and Functionals with Uniform Sublevel Sets

In this paper, we study Gerstewitz functionals that are defined on an arbitrary linear space without assuming any topology. Extended real‐valued functions with uniform sublevel sets turn out to be Gerstewitz functionals if the sublevel sets can be described by a linear shift of a set in a specified direction. Gerstewitz functionals can represent binary relations and thus act as a tool for scalarization. Sets, which are not necessarily convex, can be separated by Gerstewitz functionals. Conditions are given under which a Gerstewitz functional is finite‐valued, convex, positively homogeneous, subadditive, sublinear or monotone. The values of each Gerstewitz functional are connected with those of a sublinear function. It is shown that some Minkowski functionals–especially order unit norms–coincide with a Gerstewitz functional on a subset of the space.