Abstract
Abstract In the present paper we establish necessary and sufficient conditions under which two functions can be separated by a delta-convex function. This separation will be understood as a separation with respect to the partial order generated by the Lorentz cone. An application to a stability problem for delta-convexity is also given.
Highlights
A real function f defined on a convex subset of a real linear space is called a d.c. function if it is a difference of two convex functions
D.c. functions of one real variable were considered by numerous mathematicians
F (x) := (F (x), f (x)), x ∈ D, we can rewrite the inequality defining the notion of delta-convexity of the map F with a control function f by the formula
Summary
A real function f defined on a convex subset of a real linear space is called a d.c. function (or a delta-convex function) if it is a difference of two convex functions. The class of delta-convex functions is a linear space and an algebra and lattice. The separation in our paper will be considered with respect to the partial order generated by so called Lorentz cone which appears in a natural way in the context of delta-convexity and was introduced and examined in [9].
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