  In this paper various notions of convexity of real functions with respect to
Chebyshev systems defined over arbitrary subsets of the real line are
introduced. As an auxiliary notion, a concept of a relevant divided difference
and also a related lower Dinghas type derivative are also defined. The main
results of the paper offer various characterizations of the convexity notions
in terms of the nonnegativity of a generalized divided difference and the
corresponding lower Dinghas type derivative.
