Properties of functions generalized convex with respect to a WT-system

Continuity and convergence properties of functions, generalized convex with respect to a continuous weak Tchebysheff system, are investigated. It is shown that, under certain non-degeneracy assumptions on the weak Tchebysheff system, every function in its generalized convex cone is continuous, and pointwise convergent sequences of generalized convex functions are uniformly convergent on compact subsets of the domain. Further, it is proved that, with respect to a continuous Tchebysheff system, L_p-convergence to a continuous function, pointwise convergence and uniform convergence of a sequence of generalized convex functions are equivalent on compact subsets of the domain.