Quasi-concave functions and convex convergence to infinity

By a convex mode of convergence to infinity 〈Ck〉, we mean a sequence of nonempty closed convex subsets of a normed linear space X such that for each k, Ck+1 ⊆ int Ck and and a sequence 〈xn〉 is X is declared convergent to infinity with respect to 〈Ck〉 provided each Ck contains xn eventually. Positive convergence to infinity with respect to a pointed cone with nonempty interior as well as convergence to infinity in a fixed direction fit within this framework. In this paper we study the representation of convex modes of convergence to infinity by quasi-concave functions and associated remetrizations of the space.