Journal of Convex Analysis 20 (2013), No. 4, 937--946
Copyright Heldermann Verlag 2013
Separation by Convex Interpolation Families
Mihály Bessenyei
Institute of Mathematics, University of Debrecen, Pf. 12, 4010 Debrecen, Hungary
besse@science.unideb.hu
Patrícia Szokol
Institute of Mathematics, University of Debrecen, Pf. 12, 4010 Debrecen, Hungary
szokolp@science.unideb.hu
A set of continuous functions defined on an interval I is called an n-parameter Beckenbach family, if each n points of I × 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 Rn via a suitable homeomorphism. Then, the classical Helly Theorem guarantees the existence of a proper separator.
Keywords: Interpolation families, Haar and Chebyshev systems, Separation theorems, Helly's Theorem.
MSC: 26A51; 39B62, 52A20
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