Translative Sets and Functions and their Applications to Risk Measure Theory and Nonlinear Separation
nonlinear separation | coherent risk measure | isotonic Banach functional | topical function | set-valued convex function | translative function
Recently defined concepts such as nonlinear separation functionals due to Tammer and Weidner, coherent risk measures due to Delbaen et al., topical functions due to Gunawardena and Keane as well as isotonic Banach functionals coincide and can be traced back to books by Day from 1957 and Krasnosel'skij's from 1964. This paper is to get out the common background of these concepts and to give an extension to set--valued functions. Translativity with respect to one or a finite collection of elements turns out to be the key property of sets and functions in this note.