On convexity properties with respect to a Chebyshev system

The main purpose of this paper is to introduce various convexity concepts in terms of a positive Chebyshev system omega and give a systematic investigation of the relations among them. We generalize a celebrated theorem of Bernstein-Doetsch to the setting of omega-Jensen convexity. We also give sufficient conditions for the existence of discontinuous omega-Jensen affine functions. The concept of Wright convexity is extended to the setting of Chebyshev systems, as well, and it turns out to be an intermediate convexity property between omega-convexity and omega-Jensen convexity. For certain Chebyshev systems, we generalize the decomposition theorems of Wright convex and higher-order Wright convex functions obtained by C. T. Ng in 1987 and by Maksa and Pales in 2009, respectively.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/).