Abstract
The notions of quasiconvexity, Wright convexity and convexity for functions defined on a metric Abelian group are introduced. Various characterizations of such functions, the structural properties of the functions classes so obtained are established and several well-known results are extended to this new setting.
Highlights
Let X be a linear space and let t ∈ [0, 1]
The structure and properties of t-convex sets and t-quasiconvex, t-Wright convex, and t-convex functions and their generalizations have been investigated in a large number of recent papers, see e.g
In the paper [16] the question whether t-Wright convexity implies Jensen convexity was investigated and an affirmative answer was proved if t is a rational number
Summary
Let X be a linear space and let t ∈ [0, 1]. A subset D ⊆ X is termed t-convex if, for all x, y ∈ D, tx + (1 − t)y ∈ D. The structure and properties of t-convex sets and t-quasiconvex, t-Wright convex, and t-convex functions and their generalizations have been investigated in a large number of recent papers, see e.g. The following more general result about t-convexity was established by Kuhn [12]. In the paper [16] the question whether t-Wright convexity implies Jensen convexity was investigated and an affirmative answer was proved if t is a rational number. Bernstein–Doetsch-type theorems for Wright convex functions were established by Olbrys [29] and by Lewicki [14,15]. The purpose of this paper is to adopt and extend this definition to functions and to investigate the associated notions of quasiconvexity, Wright convexity and convexity. Some of our results will generalize Theorems A and B as well as the above described statements
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