On convexity properties with respect to a Chebyshev system

Abstract

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 Páles in 2009, respectively.