  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 P\'ales in 2009, respectively.

Classical Analysis and ODEs