Separation by Convex Interpolation Families

A set of continuous functions defined on an interval I is called an n-parameter Beckenbach family, if each n points of $I \times \mathbb{R}$ (with pairwise distinct first coordinates) can be interpolated by a unique element of the set. The aim of the present note is to characterize such pairs of real valued functions that can be separated by a member of a given convex Beckenbach family of order n. The key idea of the proof is to identify the family with $\mathbb{R}^n$ via a suitable homeomorphism. Then, the classical Helly Theorem guarantees the existence of a proper separator.