Small-Scale Magnitude Below One for Cyclic Two-Chunk Finite Metric Spaces

Abstract

Motivated by the small-scale viewpoint of Roff and Yoshinaga, we study finite metric spaces whose scaled copies collapse to a single point while their magnitude remembers how the collapse takes place. The limit metric space is geometrically indistinguishable from a point, but the magnitude function can detect differences in the path of collapse. We introduce a four-parameter family of cyclic two-chunk finite metric spaces, compute their magnitude explicitly, and use the formula to construct balanced examples whose small-scale magnitude is less than one. In particular, we exhibit a twelve-point finite metric space satisfying lim_{t -> 0+} Mag(tX) = 44/59 < 1. The guiding question and the terminology around the one-point property come from Leinster&#39;s magnitude of finite metric spaces and from Roff--Yoshinaga&#39;s work on