On Betweenness and Equidistance in Metric Spaces

The classic notions of betweenness and equidistance in Euclidean geometry readily generalize to the context of metric spaces. We view these notions from an axiomatic perspective, then analyze the role various axioms play when interpreted in a metric space – especially one where the metric is induced by a vector space norm. The points lying between two given points constitute the betweenness interval bracketed by those points, and the points equidistant from two given points constitute the equiset with those points as cocenters. An equiset gives rise to a division of the underlying space into two comparative nearness regions; in the case of the Euclidean plane, each such region is a half-plane bounded by the line that is the equiset. Betweenness intervals naturally engender a notion of convexity, and one focus of this investigation is the issue of when equisets and their comparative nearness regions, as well as the betweenness intervals themselves, are convex. For normed vector spaces, betweenness intervals are always convex when the dimension is at most two, but this convexity property easily fails in higher dimensions. Equisets and comparative nearness regions in a normed vector space are convex precisely when the norm arises from an inner product. This is one of several characterizations we present of normed vector space properties purely in terms of abstract betweenness, equidistance and comparative nearness.