On the flatness of digital hyperplanes

Abstract

This paper investigates the properties of digital hyperplanes of arbitrary dimension. We extend previous results that have been obtained for digital straight lines and digital planes, namely, Hung's evenness, Rosenfeld's chord, and Kim's chordal triangle property. To characterize digital hyperplanes we introduce the notion of digital flatness. We make a distinction between flatness and local flatness. The main tool we use is Helly's First Theorem, a classical result on convex sets, by means of which precise and verifiable conditions are given for the flatness of digital point sets. The main result is the proof of the equivalence of local flatness, evenness, and the chord property for certain infinite digital point sets in spaces of arbitrary dimension.